### The Mizar article:

### On the Topological Properties of Meet-Continuous Lattices

**by****Artur Kornilowicz**

- Received December 20, 1996
Copyright (c) 1996 Association of Mizar Users

- MML identifier: WAYBEL_9
- [ MML identifier index ]

environ vocabulary ORDERS_1, FUNCT_1, SEQM_3, PRE_TOPC, WAYBEL_0, SETFAM_1, SUBSET_1, TARSKI, RELAT_2, LATTICE3, LATTICES, RELAT_1, QUANTAL1, WELLORD1, YELLOW_0, CAT_1, ORDINAL2, WAYBEL_2, YELLOW_2, FINSUB_1, WELLORD2, YELLOW_1, YELLOW_6, BOOLE, PCOMPS_1, NATTRA_1, REALSET1, FINSET_1, COMPTS_1, CONNSP_2, TOPS_1, SEQ_2, WAYBEL_7, MCART_1, WAYBEL_9; notation TARSKI, XBOOLE_0, SUBSET_1, MCART_1, FINSUB_1, RELAT_1, RELSET_1, RELAT_2, FUNCT_1, FINSET_1, SETFAM_1, TOLER_1, PARTFUN1, FUNCT_2, STRUCT_0, PRE_TOPC, TOPS_1, TOPS_2, ORDERS_1, COMPTS_1, PCOMPS_1, REALSET1, GROUP_1, CONNSP_2, LATTICE3, ORDERS_3, TDLAT_3, YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_2, WAYBEL_1, GRCAT_1, YELLOW_4, WAYBEL_2, YELLOW_6; constructors FINSUB_1, TOPS_1, PCOMPS_1, CONNSP_2, ORDERS_3, WAYBEL_2, YELLOW_4, WAYBEL_1, TOLER_1, TOPS_2, YELLOW_6, TDLAT_3, GROUP_1, GRCAT_1, WAYBEL_3; clusters STRUCT_0, LATTICE3, WAYBEL_0, YELLOW_6, FUNCT_1, PRE_TOPC, RELSET_1, YELLOW_0, TDLAT_3, WAYBEL_2, YELLOW_4, FINSET_1, FINSUB_1, WAYBEL_3, FUNCT_2, XBOOLE_0; requirements BOOLE, SUBSET; definitions TARSKI, YELLOW_6, ORDERS_1, WAYBEL_0, WAYBEL_1, YELLOW_0, PRE_TOPC, WAYBEL_2, COMPTS_1, LATTICE3, TOPS_2, REALSET1, GROUP_1, STRUCT_0, XBOOLE_0; theorems BORSUK_1, COMPTS_1, CONNSP_2, FINSET_1, FUNCT_1, FUNCT_2, LATTICE3, MCART_1, ORDERS_1, ORDERS_2, PCOMPS_1, PRE_TOPC, RELAT_1, SETFAM_1, STRUCT_0, TARSKI, TMAP_1, TOPMETR, TOPS_1, TOPS_2, WAYBEL_0, WAYBEL_1, WAYBEL_2, WELLORD1, YELLOW_0, YELLOW_1, YELLOW_2, YELLOW_4, YELLOW_6, YELLOW_7, ZFMISC_1, RELSET_1, TDLAT_3, WAYBEL_8, GRCAT_1, XBOOLE_0, XBOOLE_1, RELAT_2; schemes SUPINF_2, YELLOW_0, FUNCT_2, XBOOLE_0; begin :: Preliminaries ::------------------------------------------------------------------- :: Acknowledgements: :: ================= :: :: I would like to thank Professor A. Trybulec for his help in the preparation :: of the article. ::------------------------------------------------------------------- definition let L be non empty RelStr; cluster id L -> monotone; coherence by YELLOW_2:13; end; definition let S, T be non empty RelStr, f be map of S,T; redefine attr f is antitone means :Def1: for x, y being Element of S st x <= y holds f.x >= f.y; compatibility proof thus f is antitone implies for x, y being Element of S st x <= y holds f.x >= f.y by WAYBEL_0:def 5; assume for x, y being Element of S st x <= y holds f.x >= f.y; hence for x, y being Element of S st x <= y for a, b being Element of T st a = f.x & b = f.y holds a >= b; end; end; theorem Th1: for S, T being RelStr, K, L being non empty RelStr for f being map of S, T, g being map of K, L st the RelStr of S = the RelStr of K & the RelStr of T = the RelStr of L & f = g & f is monotone holds g is monotone proof let S, T be RelStr, K, L be non empty RelStr, f be map of S, T, g be map of K, L such that A1: the RelStr of S = the RelStr of K and A2: the RelStr of T = the RelStr of L and A3: f = g and A4: f is monotone; reconsider S1 = S, T1 = T as non empty RelStr by A1,A2,STRUCT_0:def 1; reconsider f1 = f as map of S1, T1; let x, y be Element of K such that A5: x <= y; reconsider x1 = x, y1 = y as Element of S1 by A1; x1 <= y1 by A1,A5,YELLOW_0:1; then f1.x1 <= f1.y1 by A4,WAYBEL_1:def 2; hence g.x <= g.y by A2,A3,YELLOW_0:1; end; theorem Th2: for S, T being RelStr, K, L being non empty RelStr for f being map of S, T, g being map of K, L st the RelStr of S = the RelStr of K & the RelStr of T = the RelStr of L & f = g & f is antitone holds g is antitone proof let S, T be RelStr, K, L be non empty RelStr, f be map of S, T, g be map of K, L such that A1: the RelStr of S = the RelStr of K and A2: the RelStr of T = the RelStr of L and A3: f = g and A4: f is antitone; reconsider S1 = S, T1 = T as non empty RelStr by A1,A2,STRUCT_0:def 1; reconsider f1 = f as map of S1, T1; let x, y be Element of K such that A5: x <= y; reconsider x1 = x, y1 = y as Element of S1 by A1; x1 <= y1 by A1,A5,YELLOW_0:1; then f1.x1 >= f1.y1 by A4,Def1; hence g.x >= g.y by A2,A3,YELLOW_0:1; end; theorem Th3: for A, B being 1-sorted for F being Subset-Family of A, G being Subset-Family of B st the carrier of A = the carrier of B & F = G & F is_a_cover_of A holds G is_a_cover_of B proof let A, B be 1-sorted, F be Subset-Family of A, G be Subset-Family of B such that A1: the carrier of A = the carrier of B and A2: F = G and A3: F is_a_cover_of A; thus [#]B = the carrier of B by PRE_TOPC:12 .= [#]A by A1,PRE_TOPC:12 .= union G by A2,A3,PRE_TOPC:def 8; end; Lm1: for L being antisymmetric reflexive with_infima RelStr, x being Element of L holds uparrow x = {z where z is Element of L : z "/\" x = x} proof let L be antisymmetric reflexive with_infima RelStr, x be Element of L; thus uparrow x c= {z where z is Element of L : z "/\" x = x} proof let q be set; assume A1: q in uparrow x; then reconsider q1 = q as Element of L; x <= q1 by A1,WAYBEL_0:18; then q1 "/\" x = x by YELLOW_0:25; hence q in {z where z is Element of L : z "/\" x = x}; end; let q be set; assume q in {z where z is Element of L : z "/\" x = x}; then consider z being Element of L such that A2: q = z and A3: z "/\" x = x; x <= z by A3,YELLOW_0:25; hence q in uparrow x by A2,WAYBEL_0:18; end; theorem Th4: for L being antisymmetric reflexive with_suprema RelStr, x being Element of L holds uparrow x = {x} "\/" [#]L proof let L be antisymmetric reflexive with_suprema RelStr, x be Element of L; A1: {x} "\/" [#]L = {x "\/" s where s is Element of L : s in [#] L} by YELLOW_4:15; A2: [#]L = the carrier of L by PRE_TOPC:12; thus uparrow x c= {x} "\/" [#]L proof let q be set; assume A3: q in uparrow x; then reconsider q1 = q as Element of L; x <= q1 by A3,WAYBEL_0:18; then x "\/" q1 = q1 by YELLOW_0:24; hence q in {x} "\/" [#]L by A1,A2; end; let q be set; assume q in {x} "\/" [#]L; then consider z being Element of L such that A4: q = x "\/" z and z in [#]L by A1; x <= x "\/" z by YELLOW_0:22; hence q in uparrow x by A4,WAYBEL_0:18; end; Lm2: for L being antisymmetric reflexive with_suprema RelStr, x being Element of L holds downarrow x = {z where z is Element of L : z "\/" x = x} proof let L be antisymmetric reflexive with_suprema RelStr, x be Element of L; thus downarrow x c= {z where z is Element of L : z "\/" x = x} proof let q be set; assume A1: q in downarrow x; then reconsider q1 = q as Element of L; x >= q1 by A1,WAYBEL_0:17; then q1 "\/" x = x by YELLOW_0:24; hence q in {z where z is Element of L : z "\/" x = x}; end; let q be set; assume q in {z where z is Element of L : z "\/" x = x}; then consider z being Element of L such that A2: q = z and A3: z "\/" x = x; x >= z by A3,YELLOW_0:24; hence q in downarrow x by A2,WAYBEL_0:17; end; theorem Th5: for L being antisymmetric reflexive with_infima RelStr, x being Element of L holds downarrow x = {x} "/\" [#]L proof let L be antisymmetric reflexive with_infima RelStr, x be Element of L; A1: {x} "/\" [#]L = {x "/\" s where s is Element of L : s in [#] L} by YELLOW_4:42; A2: [#]L = the carrier of L by PRE_TOPC:12; thus downarrow x c= {x} "/\" [#]L proof let q be set; assume A3: q in downarrow x; then reconsider q1 = q as Element of L; x >= q1 by A3,WAYBEL_0:17; then x "/\" q1 = q1 by YELLOW_0:25; hence q in {x} "/\" [#]L by A1,A2; end; let q be set; assume q in {x} "/\" [#]L; then consider z being Element of L such that A4: q = x "/\" z and z in [#]L by A1; x "/\" z <= x by YELLOW_0:23; hence q in downarrow x by A4,WAYBEL_0:17; end; theorem for L being antisymmetric reflexive with_infima RelStr, y being Element of L holds (y"/\").:(uparrow y) = {y} proof let L be antisymmetric reflexive with_infima RelStr, y be Element of L; thus (y"/\").:(uparrow y) c= {y} proof let q be set; assume q in (y"/\").:(uparrow y); then consider a being set such that a in dom (y"/\") and A1: a in uparrow y and A2: q = (y"/\").a by FUNCT_1:def 12; reconsider a as Element of L by A1; A3: y <= a by A1,WAYBEL_0:18; q = y "/\" a by A2,WAYBEL_1:def 18 .= y by A3,YELLOW_0:25; hence q in {y} by TARSKI:def 1; end; let q be set; assume q in {y}; then A4: q = y by TARSKI:def 1; A5: dom (y"/\") = the carrier of L by FUNCT_2:def 1; y <= y; then A6: y in uparrow y by WAYBEL_0:18; y = y "/\" y by YELLOW_0:25 .= (y"/\").y by WAYBEL_1:def 18; hence q in (y"/\").:(uparrow y) by A4,A5,A6,FUNCT_1:def 12; end; theorem Th7: for L being antisymmetric reflexive with_infima RelStr, x being Element of L holds (x"/\")"{x} = uparrow x proof let L be antisymmetric reflexive with_infima RelStr, x be Element of L; thus (x"/\")"{x} c= uparrow x proof let q be set; assume A1: q in (x"/\")"{x}; then reconsider q1 = q as Element of L; A2: (x"/\").q1 in {x} by A1,FUNCT_1:def 13; x "/\" q1 = (x"/\").q1 by WAYBEL_1:def 18 .= x by A2,TARSKI:def 1; then x <= q1 by YELLOW_0:25; hence q in uparrow x by WAYBEL_0:18; end; let q be set; assume A3: q in uparrow x; then reconsider q1 = q as Element of L; A4: dom (x"/\") = the carrier of L by FUNCT_2:def 1; A5: x <= q1 by A3,WAYBEL_0:18; (x"/\").q1 = x "/\" q1 by WAYBEL_1:def 18 .= x by A5,YELLOW_0:25; then (x"/\").q1 in {x} by TARSKI:def 1; hence q in (x"/\")"{x} by A4,FUNCT_1:def 13; end; theorem Th8: for T being non empty 1-sorted, N being non empty NetStr over T holds N is_eventually_in rng the mapping of N proof let T be non empty 1-sorted, N be non empty NetStr over T; consider i being Element of N; take i; let j be Element of N such that i <= j; N.j = (the mapping of N).j by WAYBEL_0:def 8; hence N.j in rng the mapping of N by FUNCT_2:6; end; :: cf WAYBEL_2:19 Lm3: for L being non empty reflexive transitive RelStr for D being non empty directed Subset of L for n being Function of D, the carrier of L holds NetStr (#D,(the InternalRel of L)|_2 D,n#) is net of L proof let L be non empty reflexive transitive RelStr, D be non empty directed Subset of L, n be Function of D, the carrier of L; set N = NetStr (#D,(the InternalRel of L)|_2 D,n#); N is SubRelStr of L proof thus the carrier of N c= the carrier of L; thus the InternalRel of N c= the InternalRel of L by WELLORD1:15; end; then reconsider N as SubRelStr of L; N is full proof thus the InternalRel of N = (the InternalRel of L)|_2 the carrier of N; end; then reconsider N as full SubRelStr of L; N is directed proof let x, y be Element of N; assume x in [#]N & y in [#]N; reconsider x1 = x, y1 = y as Element of D; consider z1 being Element of L such that A1: z1 in D & x1 <= z1 & y1 <= z1 by WAYBEL_0:def 1; reconsider z = z1 as Element of N by A1; take z; D = [#]N by PRE_TOPC:12; hence z in [#]N; thus x <= z & y <= z by A1,YELLOW_0:61; end; then reconsider N as prenet of L; N is transitive; hence thesis; end; definition let L be non empty reflexive RelStr, D be non empty directed Subset of L, n be Function of D, the carrier of L; cluster NetStr (#D,(the InternalRel of L)|_2 D,n#) -> directed; coherence by WAYBEL_2:19; end; definition let L be non empty reflexive transitive RelStr, D be non empty directed Subset of L, n be Function of D, the carrier of L; cluster NetStr (#D,(the InternalRel of L)|_2 D,n#) -> transitive; coherence by Lm3; end; :: cf WAYBEL_2:42 theorem Th9: for L being non empty reflexive transitive RelStr st for x being Element of L, N being net of L st N is eventually-directed holds x "/\" sup N = sup ({x} "/\" rng netmap (N,L)) holds L is satisfying_MC proof let L be non empty reflexive transitive RelStr such that A1: for x being Element of L, N being net of L st N is eventually-directed holds x "/\" sup N = sup ({x} "/\" rng netmap (N,L)); let x be Element of L, D be non empty directed Subset of L; reconsider n = id D as Function of D, the carrier of L by FUNCT_2:9; set N = NetStr (#D,(the InternalRel of L)|_2 D,n#); D c= D; then A2: D = n.:D by BORSUK_1:3 .= rng n by FUNCT_2:45 .= rng netmap (N,L) by WAYBEL_0:def 7; A3: N is eventually-directed by WAYBEL_2:20; A4: Sup netmap (N,L) = Sup the mapping of N by WAYBEL_0:def 7 .= sup N by WAYBEL_2:def 1; thus x "/\" sup D = x "/\" Sup netmap (N,L) by A2,YELLOW_2:def 5 .= sup ({x} "/\" D) by A1,A2,A3,A4; end; theorem Th10: for L being non empty RelStr, a being Element of L, N being net of L holds a "/\" N is net of L proof let L be non empty RelStr, a be Element of L, N be net of L; set aN = a "/\" N; aN is transitive proof let x, y, z be Element of aN such that A1: x <= y & y <= z; reconsider x1 = x, y1 = y, z1 = z as Element of N by WAYBEL_2:22; A2: the RelStr of N = the RelStr of aN by WAYBEL_2:def 3; then x1 <= y1 & y1 <= z1 by A1,YELLOW_0:1; then x1 <= z1 by YELLOW_0:def 2; hence x <= z by A2,YELLOW_0:1; end; hence thesis; end; definition let L be non empty RelStr, x be Element of L, N be net of L; redefine func x "/\" N -> strict net of L; coherence by Th10; end; definition let L be non empty RelStr, x be Element of L, N be non empty reflexive NetStr over L; cluster x "/\" N -> reflexive; coherence proof the RelStr of N = the RelStr of x "/\" N by WAYBEL_2:def 3; then the InternalRel of x "/\" N is_reflexive_in the carrier of x "/\" N by ORDERS_1:def 4; hence thesis by ORDERS_1:def 4; end; end; definition let L be non empty RelStr, x be Element of L, N be non empty antisymmetric NetStr over L; cluster x "/\" N -> antisymmetric; coherence proof the RelStr of N = the RelStr of x "/\" N by WAYBEL_2:def 3; then the InternalRel of x "/\" N is_antisymmetric_in the carrier of x "/\" N by ORDERS_1:def 6; hence thesis by ORDERS_1:def 6; end; end; definition let L be non empty RelStr, x be Element of L, N be non empty transitive NetStr over L; cluster x "/\" N -> transitive; coherence proof the RelStr of N = the RelStr of x "/\" N by WAYBEL_2:def 3; then the InternalRel of x "/\" N is_transitive_in the carrier of x "/\" N by ORDERS_1:def 5; hence thesis by ORDERS_1:def 5; end; end; definition let L be non empty RelStr, J be set, f be Function of J,the carrier of L; cluster FinSups f -> transitive; coherence proof let x, y, z be Element of FinSups f such that A1: x <= y & y <= z; consider g being Function of Fin J, the carrier of L such that A2: for x being Element of Fin J holds g.x = sup (f.:x) & FinSups f = NetStr (# Fin J, RelIncl Fin J, g #) by WAYBEL_2:def 2; A3: InclPoset Fin J = RelStr(#Fin J, RelIncl Fin J#) by YELLOW_1:def 1; then reconsider x1 = x, y1 = y, z1 = z as Element of InclPoset Fin J by A2; x1 <= y1 & y1 <= z1 by A1,A2,A3,YELLOW_0:1; then x1 c= y1 & y1 c= z1 by YELLOW_1:3; then x1 c= z1 by XBOOLE_1:1; then x1 <= z1 by YELLOW_1:3; hence x <= z by A2,A3,YELLOW_0:1; end; end; begin :: The Operations Defined on Nets definition let L be non empty RelStr, N be NetStr over L; func inf N -> Element of L equals :Def2: Inf the mapping of N; coherence; end; definition let L be RelStr, N be NetStr over L; pred ex_sup_of N means :Def3: ex_sup_of rng the mapping of N,L; pred ex_inf_of N means :Def4: ex_inf_of rng the mapping of N,L; end; definition let L be RelStr; func L+id -> strict NetStr over L means :Def5: the RelStr of it = the RelStr of L & the mapping of it = id L; existence proof take NetStr (#the carrier of L, the InternalRel of L, id L#); thus thesis; end; uniqueness; end; definition let L be non empty RelStr; cluster L+id -> non empty; coherence proof the RelStr of L = the RelStr of L+id by Def5; hence thesis by STRUCT_0:def 1; end; end; definition let L be reflexive RelStr; cluster L+id -> reflexive; coherence proof the RelStr of L = the RelStr of L+id by Def5; then the InternalRel of L+id is_reflexive_in the carrier of L+id by ORDERS_1:def 4; hence thesis by ORDERS_1:def 4; end; end; definition let L be antisymmetric RelStr; cluster L+id -> antisymmetric; coherence proof the RelStr of L = the RelStr of L+id by Def5; then the InternalRel of L+id is_antisymmetric_in the carrier of L+id by ORDERS_1:def 6; hence thesis by ORDERS_1:def 6; end; end; definition let L be transitive RelStr; cluster L+id -> transitive; coherence proof the RelStr of L = the RelStr of L+id by Def5; then the InternalRel of L+id is_transitive_in the carrier of L+id by ORDERS_1:def 5; hence thesis by ORDERS_1:def 5; end; end; definition let L be with_suprema RelStr; cluster L+id -> directed; coherence proof A1: the RelStr of L = the RelStr of L+id by Def5; [#]L = the carrier of L by PRE_TOPC:12 .= [#](L+id) by A1,PRE_TOPC:12; hence [#](L+id) is directed by A1,WAYBEL_0:3; end; end; definition let L be directed RelStr; cluster L+id -> directed; coherence proof A1: [#]L is directed by WAYBEL_0:def 6; A2: the RelStr of L = the RelStr of L+id by Def5; [#]L = the carrier of L by PRE_TOPC:12 .= [#](L+id) by A2,PRE_TOPC:12; hence [#](L+id) is directed by A1,A2,WAYBEL_0:3; end; end; definition let L be non empty RelStr; cluster L+id -> monotone eventually-directed; coherence proof set N = L+id; thus N is monotone proof A1: netmap(N,L) = the mapping of N by WAYBEL_0:def 7 .= id L by Def5; the RelStr of N = the RelStr of L by Def5; hence netmap(L+id,L) is monotone by A1,Th1; end; for i being Element of N ex j being Element of N st for k being Element of N st j <= k holds N.i <= N.k proof let i be Element of N; take j = i; let k be Element of N such that A2: j <= k; A3: the RelStr of N = the RelStr of L by Def5; the mapping of N = id L by Def5; then (the mapping of N).i = i & (the mapping of N).k = k by A3,TMAP_1:91 ; then N.i = i & N.k = k by WAYBEL_0:def 8; hence N.i <= N.k by A2,A3,YELLOW_0:1; end; hence thesis by WAYBEL_0:11; end; end; definition let L be RelStr; func L opp+id -> strict NetStr over L means :Def6: the carrier of it = the carrier of L & the InternalRel of it = (the InternalRel of L)~ & the mapping of it = id L; existence proof take NetStr (#the carrier of L, (the InternalRel of L)~, id L#); thus thesis; end; uniqueness; end; theorem Th11: for L being RelStr holds the RelStr of L~ = the RelStr of L opp+id proof let L be RelStr; A1: L~ = RelStr (#the carrier of L, (the InternalRel of L)~#) by LATTICE3:def 5; then the InternalRel of L~ = the InternalRel of L opp+id by Def6; hence the RelStr of L~ = the RelStr of L opp+id by A1,Def6; end; definition let L be non empty RelStr; cluster L opp+id -> non empty; coherence proof the RelStr of L~ = the RelStr of L opp+id by Th11; hence thesis by STRUCT_0:def 1; end; end; definition let L be reflexive RelStr; cluster L opp+id -> reflexive; coherence proof the InternalRel of L is_reflexive_in the carrier of L by ORDERS_1:def 4; then A1: (the InternalRel of L)~ is_reflexive_in the carrier of L by ORDERS_2:91; the InternalRel of L opp+id = (the InternalRel of L)~ by Def6; hence the InternalRel of L opp+id is_reflexive_in the carrier of L opp+id by A1,Def6; end; end; definition let L be antisymmetric RelStr; cluster L opp+id -> antisymmetric; coherence proof the InternalRel of L is_antisymmetric_in the carrier of L by ORDERS_1:def 6; then A1: (the InternalRel of L)~ is_antisymmetric_in the carrier of L by ORDERS_2:93; the InternalRel of L opp+id = (the InternalRel of L)~ by Def6; then the InternalRel of L opp+id is_antisymmetric_in the carrier of L opp+id by A1,Def6; hence thesis by ORDERS_1:def 6; end; end; definition let L be transitive RelStr; cluster L opp+id -> transitive; coherence proof the InternalRel of L is_transitive_in the carrier of L by ORDERS_1:def 5; then A1: (the InternalRel of L)~ is_transitive_in the carrier of L by ORDERS_2:92; the InternalRel of L opp+id = (the InternalRel of L)~ by Def6; then the InternalRel of L opp+id is_transitive_in the carrier of L opp+id by A1,Def6; hence thesis by ORDERS_1:def 5; end; end; definition let L be with_infima RelStr; cluster L opp+id -> directed; coherence proof A1: the RelStr of L~ = the RelStr of L opp+id by Th11; reconsider A = L~ as with_suprema RelStr by YELLOW_7:16; [#]A = the carrier of A by PRE_TOPC:12 .= the carrier of L by YELLOW_6:12 .= the carrier of L opp+id by Def6 .= [#](L opp+id) by PRE_TOPC:12; hence [#](L opp+id) is directed by A1,WAYBEL_0:3; end; end; definition let L be non empty RelStr; cluster L opp+id -> antitone eventually-filtered; coherence proof set N = L opp+id; thus N is antitone proof A1: netmap(N,L) = the mapping of N by WAYBEL_0:def 7 .= id L by Def6; A2: the RelStr of L opp+id = the RelStr of L~ by Th11; reconsider f = id L as map of L~, L by YELLOW_7:39; reconsider g = f as map of L, L~ by YELLOW_7:39; A3: the RelStr of L = the RelStr of L; g is antitone by YELLOW_7:40; then f is antitone by YELLOW_7:41; hence netmap(N,L) is antitone by A1,A2,A3,Th2; end; for i being Element of N ex j being Element of N st for k being Element of N st j <= k holds N.i >= N.k proof let i be Element of N; take j = i; let k be Element of N such that A4: j <= k; reconsider i1 = i, k1 = k as Element of L by Def6; A5: (id L).i1 = i1 & (id L).k1 = k1 by TMAP_1:91; the mapping of N = id L by Def6; then A6: N.i = i & N.k = k by A5,WAYBEL_0:def 8; A7: the InternalRel of N = (the InternalRel of L)~ by Def6; [i,k] in the InternalRel of N by A4,ORDERS_1:def 9; then [k,i] in (the InternalRel of N)~ by RELAT_1:def 7; hence N.k <= N.i by A6,A7,ORDERS_1:def 9; end; hence thesis by WAYBEL_0:12; end; end; definition let L be non empty 1-sorted, N be non empty NetStr over L, i be Element of N; func N|i -> strict NetStr over L means :Def7: (for x being set holds x in the carrier of it iff ex y being Element of N st y = x & i <= y) & the InternalRel of it = (the InternalRel of N)|_2 the carrier of it & the mapping of it = (the mapping of N)|the carrier of it; existence proof defpred P[set] means ex y being Element of N st y = $1 & i <= y; consider X being set such that A1: for x being set holds x in X iff x in the carrier of N & P[x] from Separation; X c= the carrier of N proof let x be set; assume x in X; hence x in the carrier of N by A1; end; then reconsider f = (the mapping of N)|X as Function of X, the carrier of L by FUNCT_2:38; set IT = NetStr (#X, (the InternalRel of N)|_2 X, f#); take IT; thus for x being set holds x in the carrier of IT iff ex y being Element of N st y = x & i <= y by A1; thus thesis; end; uniqueness proof defpred P[set] means ex y being Element of N st y = $1 & i <= y; let A, B be strict NetStr over L such that A2: for x being set holds x in the carrier of A iff P[x] and A3: the InternalRel of A = (the InternalRel of N)|_2 the carrier of A and A4: the mapping of A = (the mapping of N)|the carrier of A and A5: for x being set holds x in the carrier of B iff P[x] and A6: the InternalRel of B = (the InternalRel of N)|_2 the carrier of B and A7: the mapping of B = (the mapping of N)|the carrier of B; the carrier of A = the carrier of B from Extensionality(A2, A5); hence thesis by A3,A4,A6,A7; end; end; theorem for L being non empty 1-sorted, N being non empty NetStr over L for i being Element of N holds the carrier of N|i = { y where y is Element of N : i <= y } proof let L be non empty 1-sorted, N be non empty NetStr over L, i be Element of N; thus the carrier of N|i c= { y where y is Element of N : i <= y } proof let q be set; assume q in the carrier of N|i; then consider y being Element of N such that A1: y = q & i <= y by Def7; thus thesis by A1; end; let q be set; assume q in { y where y is Element of N : i <= y }; then consider y being Element of N such that A2: q = y & i <= y; thus thesis by A2,Def7; end; theorem Th13: for L being non empty 1-sorted, N being non empty NetStr over L for i being Element of N holds the carrier of N|i c= the carrier of N proof let L be non empty 1-sorted, N be non empty NetStr over L, i be Element of N; let x be set; assume x in the carrier of N|i; then consider y being Element of N such that A1: y = x and i <= y by Def7; thus x in the carrier of N by A1; end; theorem Th14: for L being non empty 1-sorted, N being non empty NetStr over L for i being Element of N holds N|i is full SubNetStr of N proof let L be non empty 1-sorted, N be non empty NetStr over L, i be Element of N; A1: the InternalRel of N|i = (the InternalRel of N)|_2 the carrier of N|i by Def7; A2: N|i is SubRelStr of N proof thus the carrier of N|i c= the carrier of N by Th13; thus the InternalRel of N|i c= the InternalRel of N by A1,WELLORD1:15; end; N|i is SubNetStr of N proof thus N|i is SubRelStr of N by A2; thus the mapping of N|i = (the mapping of N)|the carrier of N|i by Def7; end; then reconsider K = N|i as SubNetStr of N; K is full proof thus K is full SubRelStr of N proof the InternalRel of K = (the InternalRel of N)|_2 the carrier of K by Def7; hence thesis by A2,YELLOW_0:def 14; end; end; hence thesis; end; definition let L be non empty 1-sorted, N be non empty reflexive NetStr over L, i be Element of N; cluster N|i -> non empty reflexive; coherence proof thus N|i is non empty proof i <= i; then i in the carrier of N|i by Def7; hence thesis by STRUCT_0:def 1; end; then reconsider A = N|i as strict non empty NetStr over L; A is reflexive proof let x be Element of A; consider y being Element of N such that A1: y = x and i <= y by Def7; A2: N|i is full SubNetStr of N by Th14; y <= y; hence x <= x by A1,A2,YELLOW_6:21; end; hence thesis; end; end; definition let L be non empty 1-sorted, N be non empty directed NetStr over L, i be Element of N; cluster N|i -> non empty; coherence proof consider z1 being Element of N such that A1: i <= z1 & i <= z1 by YELLOW_6:def 5; z1 in the carrier of N|i by A1,Def7; hence thesis by STRUCT_0:def 1; end; end; definition let L be non empty 1-sorted, N be non empty reflexive antisymmetric NetStr over L, i be Element of N; cluster N|i -> antisymmetric; coherence proof let x, y be Element of N|i such that A1: x <= y & y <= x; A2: N|i is SubNetStr of N by Th14; consider x1 being Element of N such that A3: x1 = x and i <= x1 by Def7; consider y1 being Element of N such that A4: y1 = y and i <= y1 by Def7; x1 <= y1 & y1 <= x1 by A1,A2,A3,A4,YELLOW_6:20; hence x = y by A3,A4,YELLOW_0:def 3; end; end; definition let L be non empty 1-sorted, N be non empty directed antisymmetric NetStr over L, i be Element of N; cluster N|i -> antisymmetric; coherence proof let x, y be Element of N|i such that A1: x <= y & y <= x; A2: N|i is SubNetStr of N by Th14; consider x1 being Element of N such that A3: x1 = x and i <= x1 by Def7; consider y1 being Element of N such that A4: y1 = y and i <= y1 by Def7; x1 <= y1 & y1 <= x1 by A1,A2,A3,A4,YELLOW_6:20; hence x = y by A3,A4,YELLOW_0:def 3; end; end; definition let L be non empty 1-sorted, N be non empty reflexive transitive NetStr over L, i be Element of N; cluster N|i -> transitive; coherence proof let x, y, z be Element of N|i such that A1: x <= y & y <= z; A2: N|i is full SubNetStr of N by Th14; consider x1 being Element of N such that A3: x1 = x and i <= x1 by Def7; consider y1 being Element of N such that A4: y1 = y and i <= y1 by Def7; consider z1 being Element of N such that A5: z1 = z and i <= z1 by Def7; x1 <= y1 & y1 <= z1 by A1,A2,A3,A4,A5,YELLOW_6:20; then x1 <= z1 by YELLOW_0:def 2; hence x <= z by A2,A3,A5,YELLOW_6:21; end; end; definition let L be non empty 1-sorted, N be net of L, i be Element of N; cluster N|i -> transitive directed; coherence proof thus N|i is transitive proof let x, y, z be Element of N|i such that A1: x <= y & y <= z; A2: N|i is full SubNetStr of N by Th14; consider x1 being Element of N such that A3: x1 = x and i <= x1 by Def7; consider y1 being Element of N such that A4: y1 = y and i <= y1 by Def7; consider z1 being Element of N such that A5: z1 = z and i <= z1 by Def7; x1 <= y1 & y1 <= z1 by A1,A2,A3,A4,A5,YELLOW_6:20; then x1 <= z1 by YELLOW_0:def 2; hence x <= z by A2,A3,A5,YELLOW_6:21; end; for x, y being Element of N|i ex z being Element of N|i st x <= z & y <= z proof let x, y be Element of N|i; consider x1 being Element of N such that A6: x1 = x and A7: i <= x1 by Def7; consider y1 being Element of N such that A8: y1 = y and i <= y1 by Def7; consider z1 being Element of N such that A9: x1 <= z1 & y1 <= z1 by YELLOW_6:def 5; i <= z1 by A7,A9,YELLOW_0:def 2; then z1 in the carrier of N|i by Def7; then reconsider z = z1 as Element of N|i; take z; N|i is full SubNetStr of N by Th14; hence x <= z & y <= z by A6,A8,A9,YELLOW_6:21; end; hence thesis by YELLOW_6:def 5; end; end; theorem for L being non empty 1-sorted, N being non empty reflexive NetStr over L for i, x being Element of N, x1 being Element of N|i st x = x1 holds N.x = (N|i).x1 proof let L be non empty 1-sorted, N be non empty reflexive NetStr over L, i, x be Element of N, x1 be Element of N|i such that A1: x = x1; thus N.x = (the mapping of N).x1 by A1,WAYBEL_0:def 8 .= ((the mapping of N)|(the carrier of N|i)).x1 by FUNCT_1:72 .= (the mapping of N|i).x1 by Def7 .= (N|i).x1 by WAYBEL_0:def 8; end; theorem Th16: for L being non empty 1-sorted, N being non empty directed NetStr over L for i, x being Element of N, x1 being Element of N|i st x = x1 holds N.x = (N|i).x1 proof let L be non empty 1-sorted, N be non empty directed NetStr over L, i, x be Element of N, x1 be Element of N|i such that A1: x = x1; thus N.x = (the mapping of N).x1 by A1,WAYBEL_0:def 8 .= ((the mapping of N)|(the carrier of N|i)).x1 by FUNCT_1:72 .= (the mapping of N|i).x1 by Def7 .= (N|i).x1 by WAYBEL_0:def 8; end; theorem Th17: for L being non empty 1-sorted, N being net of L, i being Element of N holds N|i is subnet of N proof let L be non empty 1-sorted, N be net of L, i be Element of N; reconsider A = N|i as net of L; A1: the carrier of A c= the carrier of N by Th13; A is subnet of N proof deffunc F(set) = $1; A2: for a being Element of A holds F(a) in the carrier of N by A1,TARSKI:def 3; consider f being Function of the carrier of A, the carrier of N such that A3: for x being Element of A holds f.x = F(x) from FunctR_ealEx(A2); reconsider f as map of A, N; take f; for x being set st x in the carrier of A holds (the mapping of A).x = ((the mapping of N)*f).x proof let x be set such that A4: x in the carrier of A; thus (the mapping of A).x = ((the mapping of N)|the carrier of A).x by Def7 .= (the mapping of N).x by A4,FUNCT_1:72 .= (the mapping of N).(f.x) by A3,A4 .= ((the mapping of N)*f).x by A4,FUNCT_2:21; end; hence the mapping of A = (the mapping of N)*f by FUNCT_2:18; let m be Element of N; consider z being Element of N such that A5: i <= z and A6: m <= z by YELLOW_6:def 5; z in the carrier of A by A5,Def7; then reconsider n = z as Element of A; take n; let p be Element of A such that A7: n <= p; p in the carrier of A; then reconsider p1 = p as Element of N by A1; A is SubNetStr of N by Th14; then z <= p1 by A7,YELLOW_6:20; then m <= p1 by A6,YELLOW_0:def 2; hence m <= f.p by A3; end; hence thesis; end; definition let T be non empty 1-sorted, N be net of T; cluster strict subnet of N; existence proof set A = NetStr (#the carrier of N,the InternalRel of N,the mapping of N#); A1: the RelStr of A = the RelStr of N; A is directed proof A2: [#]N is directed by WAYBEL_0:def 6; [#]N = the carrier of N by PRE_TOPC:12 .= [#]A by PRE_TOPC:12; hence [#]A is directed by A1,A2,WAYBEL_0:3; end; then reconsider A as net of T by A1,WAYBEL_8:13; A is subnet of N proof reconsider f = id N as map of A, N; take f; thus the mapping of A = (the mapping of N)*f by TMAP_1:93; let m be Element of N; reconsider n = m as Element of A; take n; let p be Element of A such that A3: n <= p; p = f.p by TMAP_1:91; hence m <= f.p by A1,A3,YELLOW_0:1; end; then reconsider A as subnet of N; take A; thus thesis; end; end; definition let L be non empty 1-sorted, N be net of L, i be Element of N; redefine func N|i -> strict subnet of N; coherence by Th17; end; definition let S be non empty 1-sorted, T be 1-sorted, f be map of S, T, N be NetStr over S; func f * N -> strict NetStr over T means :Def8: the RelStr of it = the RelStr of N & the mapping of it = f * the mapping of N; existence proof take NetStr (#the carrier of N, the InternalRel of N, f*the mapping of N#); thus thesis; end; uniqueness; end; definition let S be non empty 1-sorted, T be 1-sorted, f be map of S, T, N be non empty NetStr over S; cluster f * N -> non empty; coherence proof the RelStr of N = the RelStr of f * N by Def8; hence thesis by STRUCT_0:def 1; end; end; definition let S be non empty 1-sorted, T be 1-sorted, f be map of S, T, N be reflexive NetStr over S; cluster f * N -> reflexive; coherence proof the RelStr of N = the RelStr of f * N by Def8; hence the InternalRel of f*N is_reflexive_in the carrier of f*N by ORDERS_1:def 4; end; end; definition let S be non empty 1-sorted, T be 1-sorted, f be map of S, T, N be antisymmetric NetStr over S; cluster f * N -> antisymmetric; coherence proof the RelStr of N = the RelStr of f * N by Def8; then the InternalRel of f*N is_antisymmetric_in the carrier of f*N by ORDERS_1:def 6; hence thesis by ORDERS_1:def 6; end; end; definition let S be non empty 1-sorted, T be 1-sorted, f be map of S, T, N be transitive NetStr over S; cluster f * N -> transitive; coherence proof the RelStr of N = the RelStr of f * N by Def8; then the InternalRel of f*N is_transitive_in the carrier of f*N by ORDERS_1:def 5; hence thesis by ORDERS_1:def 5; end; end; definition let S be non empty 1-sorted, T be 1-sorted, f be map of S, T, N be directed NetStr over S; cluster f * N -> directed; coherence proof A1: the RelStr of N = the RelStr of f * N by Def8; thus [#](f*N) is directed proof A2: [#]N is directed by WAYBEL_0:def 6; [#](f*N) = the carrier of f*N by PRE_TOPC:12 .= [#]N by A1,PRE_TOPC:12; hence thesis by A1,A2,WAYBEL_0:3; end; end; end; theorem Th18: for L being non empty RelStr, N being non empty NetStr over L for x being Element of L holds (x"/\")*N = x "/\" N proof let L be non empty RelStr, N be non empty NetStr over L, x be Element of L; A1: the RelStr of (x"/\")*N = the RelStr of N by Def8 .= the RelStr of x "/\" N by WAYBEL_2:def 3; set n = the mapping of N; A2: the RelStr of N = the RelStr of x "/\" N by WAYBEL_2:def 3; then reconsider f2 = the mapping of x "/\" N as Function of the carrier of N, the carrier of L; A3: for c being Element of N holds ((x"/\") * n).c = f2.c proof let c be Element of N; consider y being Element of L such that A4: y = n.c and A5: f2.c = x "/\" y by A2,WAYBEL_2:def 3; thus ((x"/\") * n).c = (x"/\").y by A4,FUNCT_2:21 .= f2.c by A5,WAYBEL_1:def 18; end; the mapping of (x"/\")*N = (x"/\") * n by Def8 .= the mapping of x "/\" N by A3,FUNCT_2:113; hence (x"/\")*N = x "/\" N by A1; end; begin :: The Properties of Topological Spaces theorem Th19: for S, T being TopStruct for F being Subset-Family of S, G being Subset-Family of T st the TopStruct of S = the TopStruct of T & F = G & F is open holds G is open proof let S, T be TopStruct, F be Subset-Family of S, G be Subset-Family of T such that A1: the TopStruct of S = the TopStruct of T and A2: F = G and A3: F is open; let P be Subset of T such that A4: P in G; reconsider R = P as Subset of S by A1; R is open by A2,A3,A4,TOPS_2:def 1; hence P in the topology of T by A1,PRE_TOPC:def 5; end; theorem for S, T being TopStruct for F being Subset-Family of S, G being Subset-Family of T st the TopStruct of S = the TopStruct of T & F = G & F is closed holds G is closed proof let S, T be TopStruct, F be Subset-Family of S, G be Subset-Family of T such that A1: the TopStruct of S = the TopStruct of T and A2: F = G and A3: F is closed; let P be Subset of T such that A4: P in G; reconsider R = P as Subset of S by A1; R is closed by A2,A3,A4,TOPS_2:def 2; then [#]S \ R is open by PRE_TOPC:def 6; then [#]S \ R in the topology of S by PRE_TOPC:def 5; then (the carrier of S) \ R in the topology of S by PRE_TOPC:12; hence [#]T \ P in the topology of T by A1,PRE_TOPC:12; end; definition struct(TopStruct,RelStr) TopRelStr (# carrier -> set, InternalRel -> (Relation of the carrier), topology -> Subset-Family of the carrier #); end; definition let A be non empty set, R be Relation of A,A, T be Subset-Family of A; cluster TopRelStr (#A,R,T#) -> non empty; coherence proof thus the carrier of TopRelStr (#A,R,T#) is non empty; end; end; definition let x be set, R be Relation of {x}; let T be Subset-Family of {x}; cluster TopRelStr (#{x}, R, T#) -> trivial; coherence proof let a, b be Element of TopRelStr (#{x},R,T#); a = x & b = x by TARSKI:def 1; hence thesis; end; end; definition let X be set, O be Order of X, T be Subset-Family of X; cluster TopRelStr (#X, O, T#) -> reflexive transitive antisymmetric; coherence proof set A = TopRelStr(#X,O,T#); field the InternalRel of A = the carrier of A by ORDERS_1:97; then the InternalRel of A is_reflexive_in the carrier of A & the InternalRel of A is_transitive_in the carrier of A & the InternalRel of A is_antisymmetric_in the carrier of A by RELAT_2:def 9,def 12,def 16; hence thesis by ORDERS_1:def 4,def 5,def 6; end; end; Lm4: for tau being Subset-Family of {0}, r being Relation of {0} st tau = {{},{0}} & r = {[0,0]} holds TopRelStr (#{0},r,tau#) is trivial reflexive non empty discrete finite proof let tau be Subset-Family of {0}, r be Relation of {0} such that A1: tau = {{},{0}} and A2: r = {[0,0]}; set T = TopRelStr (#{0},r,tau#); thus T is trivial; thus T is reflexive proof let x be Element of T; x = 0 by TARSKI:def 1; then [x,x] in {[0,0]} by TARSKI:def 1; hence x <= x by A2,ORDERS_1:def 9; end; thus T is non empty; the topology of T = bool the carrier of T by A1,ZFMISC_1:30; hence T is discrete by TDLAT_3:def 1; thus the carrier of T is finite; end; definition cluster trivial reflexive non empty discrete strict finite TopRelStr; existence proof {{},{0}} = bool {0} & bool {0} c= bool {0} by ZFMISC_1:30; then reconsider tau = {{},{0}} as Subset-Family of {0}; 0 in {0} by TARSKI:def 1; then reconsider r = {[0,0]} as Relation of {0} by RELSET_1:8; take TopRelStr (#{0},r,tau#); thus thesis by Lm4; end; end; definition mode TopLattice is with_infima with_suprema reflexive transitive antisymmetric TopSpace-like TopRelStr; end; definition cluster strict non empty trivial discrete finite compact Hausdorff TopLattice; existence proof A1: {{},{0}} = bool {0} & bool {0} c= bool {0} by ZFMISC_1:30; 0 in {0} by TARSKI:def 1; then reconsider r = {[0,0]} as Relation of {0} by RELSET_1:8; A2: 1TopSp {0} = TopStruct (# {0}, bool {0}#) by PCOMPS_1:def 1; reconsider C = bool {0} as Subset-Family of {0}; reconsider A = TopRelStr (#{0}, r, C#) as trivial reflexive non empty discrete finite TopRelStr by A1,Lm4; reconsider A as TopLattice; take A; thus A is strict non empty trivial discrete finite; thus A is compact proof let F be Subset-Family of A such that A3: F is_a_cover_of A and A4: F is open; reconsider F1 = F as Subset-Family of 1TopSp {0} by A2; A5: F1 is_a_cover_of 1TopSp {0} by A2,A3,Th3; A6: F1 is open by A2,A4,Th19; 1TopSp {0} is compact by PCOMPS_1:9; then consider G1 being Subset-Family of 1TopSp {0} such that A7: G1 c= F1 and A8: G1 is_a_cover_of 1TopSp {0} and A9: G1 is finite by A5,A6,COMPTS_1:def 3; reconsider G = G1 as Subset-Family of A by A2; take G; thus G c= F by A7; thus G is_a_cover_of A by A2,A8,Th3; thus G is finite by A9; end; let p, q be Point of A such that A10: not p = q; assume not ex W, V being Subset of A st W is open & V is open & p in W & q in V & W misses V; p = 0 & q = 0 by TARSKI:def 1; hence contradiction by A10; end; end; definition let T be Hausdorff (non empty TopSpace); cluster -> Hausdorff (non empty SubSpace of T); coherence by TOPMETR:3; end; theorem Th21: for T being non empty TopSpace, p being Point of T for A being Element of OpenNeighborhoods p holds A is a_neighborhood of p proof let T be non empty TopSpace, p be Point of T, A be Element of OpenNeighborhoods p; consider W being Subset of T such that A1: W = A & p in W & W is open by YELLOW_6:38; thus A is a_neighborhood of p by A1,CONNSP_2:5; end; theorem Th22: for T being non empty TopSpace, p being Point of T for A, B being Element of OpenNeighborhoods p holds A /\ B is Element of OpenNeighborhoods p proof let T be non empty TopSpace, p be Point of T, A, B be Element of OpenNeighborhoods p; consider W being Subset of T such that A1: W = A & p in W & W is open by YELLOW_6:38; consider X being Subset of T such that A2: X = B & p in X & X is open by YELLOW_6:38; A3: p in A /\ B by A1,A2,XBOOLE_0:def 3; W /\ X is open by A1,A2,TOPS_1:38; then A /\ B in the carrier of OpenNeighborhoods p by A1,A2,A3,YELLOW_6:39; hence A /\ B is Element of OpenNeighborhoods p; end; theorem for T being non empty TopSpace, p being Point of T for A, B being Element of OpenNeighborhoods p holds A \/ B is Element of OpenNeighborhoods p proof let T be non empty TopSpace, p be Point of T, A, B be Element of OpenNeighborhoods p; consider W being Subset of T such that A1: W = A & p in W & W is open by YELLOW_6:38; consider X being Subset of T such that A2: X = B & p in X & X is open by YELLOW_6:38; A3: p in A \/ B by A1,XBOOLE_0:def 2; W \/ X is open by A1,A2,TOPS_1:37; then A \/ B in the carrier of OpenNeighborhoods p by A1,A2,A3,YELLOW_6:39; hence A \/ B is Element of OpenNeighborhoods p; end; theorem Th24: for T being non empty TopSpace, p being Element of T for N being net of T st p in Lim N for S being Subset of T st S = rng the mapping of N holds p in Cl S proof let T be non empty TopSpace, p be Element of T, N be net of T such that A1: p in Lim N; let S be Subset of T; assume S = rng the mapping of N; then A2: N is_eventually_in S by Th8; for G being Subset of T st G is open holds p in G implies S meets G proof let G be Subset of T such that A3: G is open and A4: p in G; G = Int G by A3,TOPS_1:55; then reconsider V = G as a_neighborhood of p by A4,CONNSP_2:def 1; N is_eventually_in V by A1,YELLOW_6:def 18; hence S meets G by A2,YELLOW_6:26; end; hence p in Cl S by PRE_TOPC:def 13; end; theorem Th25: for T being Hausdorff TopLattice, N being convergent net of T for f being map of T, T st f is continuous holds f.(lim N) in Lim (f * N) proof let T be Hausdorff TopLattice, N be convergent net of T, f be map of T, T such that A1: f is continuous; for V being a_neighborhood of f.(lim N) holds f*N is_eventually_in V proof let V be a_neighborhood of f.(lim N); consider O being a_neighborhood of lim N such that A2: f.:O c= V by A1,BORSUK_1:def 2; lim N in Lim N by YELLOW_6:def 20; then N is_eventually_in O by YELLOW_6:def 18; then consider s0 being Element of N such that A3: for j being Element of N st s0 <= j holds N.j in O by WAYBEL_0:def 11; A4: the RelStr of f*N = the RelStr of N by Def8; then reconsider s1 = s0 as Element of f*N; take s1; let j1 be Element of f*N such that A5: s1 <= j1; A6: the carrier of f*N = dom the mapping of N by A4,FUNCT_2:def 1; reconsider j = j1 as Element of N by A4; A7: (f*N).j1 = (the mapping of f*N).j1 by WAYBEL_0:def 8 .= (f * the mapping of N).j1 by Def8 .= f.((the mapping of N).j1) by A6,FUNCT_1:23 .= f.(N.j) by WAYBEL_0:def 8; s0 <= j by A4,A5,YELLOW_0:1; then A8: N.j in O by A3; dom f = the carrier of T by FUNCT_2:def 1; then f.(N.j) in f.:O by A8,FUNCT_1:def 12; hence (f*N).j1 in V by A2,A7; end; hence f.(lim N) in Lim (f * N) by YELLOW_6:def 18; end; theorem Th26: for T being Hausdorff TopLattice, N being convergent net of T for x being Element of T st x"/\" is continuous holds x "/\" lim N in Lim (x "/\" N) proof let T be Hausdorff TopLattice, N be convergent net of T, x be Element of T such that A1: x"/\" is continuous; x "/\" lim N = (x"/\").(lim N) by WAYBEL_1:def 18; then x "/\" lim N in Lim (x"/\" * N) by A1,Th25; hence x "/\" lim N in Lim (x "/\" N) by Th18; end; theorem Th27: for S being Hausdorff TopLattice, x being Element of S st for a being Element of S holds a"/\" is continuous holds uparrow x is closed proof let S be Hausdorff TopLattice, x be Element of S; assume for a being Element of S holds a"/\" is continuous; then A1: x"/\" is continuous; A2: (x"/\")"{x} = uparrow x by Th7; {x} is closed by PCOMPS_1:10; hence uparrow x is closed by A1,A2,PRE_TOPC:def 12; end; theorem Th28: for S being compact Hausdorff TopLattice, x being Element of S st for b being Element of S holds b"/\" is continuous holds downarrow x is closed proof let S be compact Hausdorff TopLattice, b be Element of S such that A1: for a being Element of S holds a"/\" is continuous; A2: {b} "/\" [#]S = {b "/\" y where y is Element of S: y in [#] S} by YELLOW_4:42; A3: b"/\" is continuous by A1; set g1 = (rng (b"/\"))|(b"/\"); A4: g1 = b"/\" by RELAT_1:125; A5: dom (b"/\") = the carrier of S by FUNCT_2:def 1; then A6: dom (b"/\") = [#]S by PRE_TOPC:12; A7: rng (b"/\") = {b} "/\" [#]S proof thus rng (b"/\") c= {b} "/\" [#]S proof let q be set; assume q in rng (b"/\"); then consider x being set such that A8: x in dom (b"/\") and A9: (b"/\").x = q by FUNCT_1:def 5; reconsider x1 = x as Element of S by A8; q = b "/\" x1 by A9,WAYBEL_1:def 18; hence q in {b} "/\" [#]S by A2,A5,A6; end; let q be set; assume q in {b} "/\" [#]S; then consider y being Element of S such that A10: q = b "/\" y & y in [#]S by A2; q = (b"/\").y by A10,WAYBEL_1:def 18; hence q in rng (b"/\") by A5,FUNCT_1:def 5; end; then rng g1 = {b} "/\" [#]S by RELAT_1:125 .= [#](S | (rng (b"/\"))) by A7,PRE_TOPC:def 10 .= the carrier of S|(rng(b"/\")) by PRE_TOPC:12; then g1 is Function of the carrier of S, the carrier of S|(rng(b"/\")) by A4,A5,FUNCT_2:3; then reconsider g1 as map of S, S|(rng (b"/\")); A11: rng g1 = {b} "/\" [#]S by A7,RELAT_1:125 .= [#](S | ({b}"/\"[#]S)) by PRE_TOPC:def 10; g1 is continuous by A3,A4,TOPMETR:9; then S | ({b} "/\" [#]S) is compact by A7,A11,COMPTS_1:23; then {b} "/\" [#]S is compact by COMPTS_1:12; then {b} "/\" [#]S is closed by COMPTS_1:16; hence downarrow b is closed by Th5; end; begin :: The Cluster Points of Nets definition let T be non empty TopSpace, N be non empty NetStr over T, p be Point of T; pred p is_a_cluster_point_of N means :Def9: for O being a_neighborhood of p holds N is_often_in O; end; theorem for L being non empty TopSpace, N being net of L for c being Point of L st c in Lim N holds c is_a_cluster_point_of N proof let L be non empty TopSpace, N be net of L, c be Point of L such that A1: c in Lim N; let O be a_neighborhood of c; N is_eventually_in O by A1,YELLOW_6:def 18; hence N is_often_in O by YELLOW_6:28; end; theorem Th30: for T being compact Hausdorff (non empty TopSpace), N being net of T ex c being Point of T st c is_a_cluster_point_of N proof let T be compact Hausdorff (non empty TopSpace), N be net of T; defpred P[set,set] means ex X being Subset of T, a being Element of N st a = $1 & X = {N.j where j is Element of N : a <= j} & $2 = Cl X; A1: for e being Element of N ex u being Element of bool the carrier of T st P[e,u] proof let e be Element of N; reconsider a = e as Element of N; {N.j where j is Element of N : a <= j} c= the carrier of T proof let q be set; assume q in {N.j where j is Element of N : a <= j}; then consider j being Element of N such that A2: q = N.j & a <= j; thus q in the carrier of T by A2; end; then reconsider X = {N.j where j is Element of N : a <= j} as Subset of the carrier of T; take Cl X, X, a; thus thesis; end; consider F being Function of the carrier of N, bool the carrier of T such that A3: for e being Element of N holds P[e,F.e] from FuncExD(A1); A4: dom F = the carrier of N by FUNCT_2:def 1; rng F c= bool the carrier of T by RELSET_1:12; then rng F is Subset-Family of T by SETFAM_1:def 7; then reconsider RF = rng F as Subset-Family of T; A5: RF is centered proof thus RF <> {} by A4,RELAT_1:65; let H be set such that A6: H <> {} and A7: H c= RF and A8: H is finite; reconsider H1 = H as non empty set by A6; set J = {i where i is Element of N : ex h, Ch being Subset of T st h = {N.j where j is Element of N : i <= j} & Ch = Cl h}; consider e being Element of H1; e in RF by A7,TARSKI:def 3; then consider x being set such that A9: x in dom F and e = F.x by FUNCT_1:def 5; reconsider x as Element of N by A9; consider X being Subset of T, a being Element of N such that a = x and A10: X = {N.j where j is Element of N : a <= j} & F.x = Cl X by A3; a in J by A10; then reconsider J as non empty set; defpred P[set,set] means ex i being Element of N, h, Ch being Subset of T st i = $2 & Ch = $1 & h = {N.j where j is Element of N : i <= j} & Ch = Cl h; A11: for e being Element of H1 ex u being Element of J st P[e,u] proof let e be Element of H1; e in RF by A7,TARSKI:def 3; then consider x being set such that A12: x in dom F and A13: e = F.x by FUNCT_1:def 5; reconsider x as Element of N by A12; consider X being Subset of T, a being Element of N such that A14: a = x and A15: X = {N.j where j is Element of N : a <= j} and A16: F.x = Cl X by A3; a in J by A15,A16; then reconsider a as Element of J; take u = a, i = x, h = X, Ch = Cl X; thus i = u by A14; thus Ch = e by A13,A16; thus h = {N.j where j is Element of N : i <= j} by A14,A15; thus Ch = Cl h; end; consider Q being Function of H1, J such that A17: for e being Element of H1 holds P[e,Q.e] from FuncExD(A11); dom Q = H by FUNCT_2:def 1; then A18: rng Q is finite by A8,FINSET_1:26; rng Q c= [#]N proof let q be set; assume q in rng Q; then consider x being set such that A19: x in dom Q and A20: Q.x = q by FUNCT_1:def 5; reconsider x as Element of H1 by A19; consider i being Element of N, h, Ch being Subset of T such that A21: i = Q.x and Ch = x & h = {N.j where j is Element of N : i <= j} & Ch = Cl h by A17; thus q in [#]N by A20,A21,PRE_TOPC:13; end; then reconsider RQ = rng Q as Subset of [#]N; [#]N is non empty directed by WAYBEL_0:def 6; then consider i0 being Element of N such that i0 in [#]N and A22: i0 is_>=_than RQ by A18,WAYBEL_0:1; for Y being set holds Y in H implies N.i0 in Y proof let Y be set; assume A23: Y in H; then A24: Y in dom Q by FUNCT_2:def 1; consider i being Element of N, h, Ch being Subset of T such that A25: i = Q.Y and A26: Ch = Y and A27: h = {N.j where j is Element of N : i <= j} and A28: Ch = Cl h by A17,A23; i in rng Q by A24,A25,FUNCT_1:def 5; then i <= i0 by A22,LATTICE3:def 9; then A29: N.i0 in h by A27; h c= Cl h by PRE_TOPC:48; hence N.i0 in Y by A26,A28,A29; end; hence meet H <> {} by A6,SETFAM_1:def 1; end; RF is closed proof let P be Subset of T; assume P in RF; then consider x being set such that A30: x in dom F and A31: F.x = P by FUNCT_1:def 5; reconsider x as Element of N by A30; consider X being Subset of T, a being Element of N such that a = x & X = {N.j where j is Element of N : a <= j} and A32: F.x = Cl X by A3; P = Cl P by A31,A32,TOPS_1:26; hence P is closed by PRE_TOPC:52; end; then meet RF <> {} by A5,COMPTS_1:13; then consider c being set such that A33: c in meet RF by XBOOLE_0:def 1; reconsider c as Point of T by A33; take c; assume not c is_a_cluster_point_of N; then consider O being a_neighborhood of c such that A34: not N is_often_in O by Def9; N is_eventually_in (the carrier of T) \ O by A34,WAYBEL_0:10; then consider s0 being Element of N such that A35: for j being Element of N st s0 <= j holds N.j in (the carrier of T) \ O by WAYBEL_0:def 11; consider Y being Subset of T, a being Element of N such that A36: a = s0 and A37: Y = {N.j where j is Element of N : a <= j} and A38: F.s0 = Cl Y by A3; Cl Y in RF by A4,A38,FUNCT_1:def 5; then A39: c in Cl Y by A33,SETFAM_1:def 1; A40: Int O c= O by TOPS_1:44; {} = O /\ Y proof thus {} c= O /\ Y by XBOOLE_1:2; let q be set such that A41: q in O /\ Y; assume not q in {}; q in Y by A41,XBOOLE_0:def 3; then consider j being Element of N such that A42: q = N.j and A43: s0 <= j by A36,A37; N.j in (the carrier of T) \ O by A35,A43; then not N.j in O by XBOOLE_0:def 4; hence contradiction by A41,A42,XBOOLE_0:def 3; end; then O misses Y by XBOOLE_0:def 7; then A44: Int O misses Y by A40,XBOOLE_1:63; A45: Int O is open by TOPS_1:51; c in Int O by CONNSP_2:def 1; hence contradiction by A39,A44,A45,PRE_TOPC:def 13; end; theorem Th31: for L being non empty TopSpace, N being net of L, M being subnet of N for c being Point of L st c is_a_cluster_point_of M holds c is_a_cluster_point_of N proof let L be non empty TopSpace, N be net of L, M be subnet of N, c be Point of L such that A1: c is_a_cluster_point_of M; let O be a_neighborhood of c; M is_often_in O by A1,Def9; hence N is_often_in O by YELLOW_6:27; end; theorem Th32: for T being non empty TopSpace, N being net of T for x being Point of T st x is_a_cluster_point_of N holds ex M being subnet of N st x in Lim M proof let T be non empty TopSpace, N be net of T, x be Point of T such that A1: x is_a_cluster_point_of N; set n = the mapping of N; set S' = {[s,O] where s is Element of N, O is Element of OpenNeighborhoods x : N.s in O}; A2: x in [#]T by PRE_TOPC:13; [#]T is open by TOPS_1:53; then [#]T in the carrier of OpenNeighborhoods x by A2,YELLOW_6:39; then reconsider O = [#]T as Element of OpenNeighborhoods x; consider q being Element of N; N.q in [#]T by PRE_TOPC:13; then [q,O] in S'; then reconsider S' as non empty set; defpred P[set,set] means ex s1, s2 being Element of N st s1 = $1`1 & s2 = $2`1 & s1 <= s2 & $2`2 c= $1`2; consider L being non empty strict RelStr such that A3: the carrier of L = S' and A4: for a, b being Element of L holds a <= b iff P[a,b] from RelStrEx; deffunc F(Element of L) = n.($1`1); A5: for a being Element of L holds F(a) in the carrier of T proof let a be Element of L; a in S' by A3; then consider s being Element of N, O being Element of OpenNeighborhoods x such that A6: a = [s,O] and N.s in O; n.(a`1) = n.s by A6,MCART_1:7; hence n.(a`1) in the carrier of T; end; consider g being Function of the carrier of L, the carrier of T such that A7: for x being Element of L holds g.x = F(x) from FunctR_ealEx(A5); set M = NetStr (#the carrier of L, the InternalRel of L, g#); for a, b being Element of M ex z being Element of M st a <= z & b <= z proof let a, b be Element of M; a in S' by A3; then consider aa being Element of N, Oa being Element of OpenNeighborhoods x such that A8: a = [aa,Oa] and N.aa in Oa; b in S' by A3; then consider bb being Element of N, Ob being Element of OpenNeighborhoods x such that A9: b = [bb,Ob] and N.bb in Ob; consider z1 being Element of N such that A10: aa <= z1 & bb <= z1 by YELLOW_6:def 5; Oa is a_neighborhood of x & Ob is a_neighborhood of x by Th21; then Oa /\ Ob is a_neighborhood of x by CONNSP_2:4; then N is_often_in Oa /\ Ob by A1,Def9; then consider d being Element of N such that A11: z1 <= d and A12: N.d in Oa /\ Ob by WAYBEL_0:def 12; Oa /\ Ob is Element of OpenNeighborhoods x by Th22; then [d, Oa /\ Ob] in S' by A12; then reconsider z = [d, Oa /\ Ob] as Element of M by A3; take z; reconsider A1 = a, C7 = b, z2 = z as Element of L; A13: Oa /\ Ob c= Oa & Oa /\ Ob c= Ob by XBOOLE_1:17; A14: aa <= d & bb <= d by A10,A11,YELLOW_0:def 2; A1`1 = aa & C7`1 = bb & z2`1 = d & A1`2 = Oa & C7`2 = Ob & z2`2 = Oa /\ Ob by A8,A9,MCART_1:7; then A1 <= z2 & C7 <= z2 by A4,A13,A14; hence a <= z & b <= z by YELLOW_0:1; end; then reconsider M as prenet of T by YELLOW_6:def 5; M is transitive proof let x, y, z be Element of M such that A15: x <= y and A16: y <= z; reconsider x1 = x, y1 = y, z1 = z as Element of L; x1 <= y1 by A15,YELLOW_0:1; then consider sx, sy being Element of N such that A17: sx = x1`1 & sy = y1`1 & sx <= sy & y1`2 c= x1`2 by A4; y1 <= z1 by A16,YELLOW_0:1; then consider s1, s2 being Element of N such that A18: s1 = y1`1 & s2 = z1`1 & s1 <= s2 & z1`2 c= y1`2 by A4; A19: sx <= s2 by A17,A18,YELLOW_0:def 2; z1`2 c= x1`2 by A17,A18,XBOOLE_1:1; then x1 <= z1 by A4,A17,A18,A19; hence x <= z by YELLOW_0:1; end; then reconsider M as net of T; M is subnet of N proof deffunc F(Element of L) = $1`1; A20: for a being Element of L holds F(a) in the carrier of N proof let a be Element of L; a in S' by A3; then consider s being Element of N, O being Element of OpenNeighborhoods x such that A21: a = [s,O] and N.s in O; a`1 = s by A21,MCART_1:7; hence a`1 in the carrier of N; end; consider f being Function of the carrier of L, the carrier of N such that A22: for x being Element of L holds f.x = F(x) from FunctR_ealEx(A20); reconsider f as map of M, N; take f; for x being set st x in the carrier of M holds g.x = (n*f).x proof let x be set; assume A23: x in the carrier of M; hence g.x = n.(x`1) by A7 .= n.(f.x) by A22,A23 .= (n*f).x by A23,FUNCT_2:21; end; hence the mapping of M = (the mapping of N)*f by FUNCT_2:18; let m be Element of N; N.m in [#]T by PRE_TOPC:13; then [m,O] in S'; then reconsider n = [m,O] as Element of M by A3; take n; let p be Element of M such that A24: n <= p; reconsider n1 = n, p1 = p as Element of L; n1 <= p1 by A24,YELLOW_0:1; then consider s1, s2 being Element of N such that A25: s1 = n1`1 & s2 = p1`1 & s1 <= s2 and p1`2 c= n1`2 by A4; f.p = p`1 by A22; hence m <= f.p by A25,MCART_1:7; end; then reconsider M as subnet of N; take M; for V being a_neighborhood of x holds M is_eventually_in V proof let V be a_neighborhood of x; consider i being Element of N; A26: x in Int V by CONNSP_2:def 1; Int V is open by TOPS_1:51; then Int V in the carrier of OpenNeighborhoods x by A26,YELLOW_6:39; then A27: Int V is Element of OpenNeighborhoods x; then Int V is a_neighborhood of x by Th21; then N is_often_in Int V by A1,Def9; then consider s being Element of N such that i <= s and A28: N.s in Int V by WAYBEL_0:def 12; A29: M is_eventually_in Int V proof [s,Int V] in S' by A27,A28; then reconsider j = [s,Int V] as Element of M by A3; take j; let s' be Element of M such that A30: j <= s'; s' in S' by A3; then consider ss being Element of N, OO being Element of OpenNeighborhoods x such that A31: s' = [ss,OO] and A32: N.ss in OO; reconsider j1 = j, s1 = s' as Element of L; j1 <= s1 by A30,YELLOW_0:1; then consider s1, s2 being Element of N such that s1 = j`1 & s2 = s'`1 & s1 <= s2 and A33: s'`2 c= j`2 by A4; A34: j`2 = Int V by MCART_1:7; A35: s'`2 = OO by A31,MCART_1:7; M.s' = (the mapping of M).s' by WAYBEL_0:def 8 .= n.(s'`1) by A7 .= n.ss by A31,MCART_1:7 .= N.ss by WAYBEL_0:def 8; hence M.s' in Int V by A32,A33,A34,A35; end; Int V c= V by TOPS_1:44; hence M is_eventually_in V by A29,WAYBEL_0:8; end; hence x in Lim M by YELLOW_6:def 18; end; theorem Th33: for L being compact Hausdorff (non empty TopSpace), N being net of L st for c, d being Point of L st c is_a_cluster_point_of N & d is_a_cluster_point_of N holds c = d holds for s being Point of L st s is_a_cluster_point_of N holds s in Lim N proof let L be compact Hausdorff (non empty TopSpace), N be net of L such that A1: for c, d being Point of L st c is_a_cluster_point_of N & d is_a_cluster_point_of N holds c = d; let c be Point of L such that A2: c is_a_cluster_point_of N; assume not c in Lim N; then consider M being subnet of N such that A3: not ex P being subnet of M st c in Lim P by YELLOW_6:46; consider d being Point of L such that A4: d is_a_cluster_point_of M by Th30; A5: c <> d by A3,A4,Th32; d is_a_cluster_point_of N by A4,Th31; hence contradiction by A1,A2,A5; end; theorem Th34: for S being non empty TopSpace, c being Point of S for N being net of S, A being Subset of S st c is_a_cluster_point_of N & A is closed & rng the mapping of N c= A holds c in A proof let S be non empty TopSpace, c be Point of S, N be net of S, A be Subset of S such that A1: c is_a_cluster_point_of N and A2: A is closed and A3: rng the mapping of N c= A; consider M being subnet of N such that A4: c in Lim M by A1,Th32; consider f being map of M, N such that A5: the mapping of M = (the mapping of N)*f and for m being Element of N ex n being Element of M st for p being Element of M st n <= p holds m <= f.p by YELLOW_6:def 12; reconsider R = rng the mapping of M as Subset of S; R c= rng the mapping of N by A5,RELAT_1:45; then R c= A by A3,XBOOLE_1:1; then A6: Cl R c= Cl A by PRE_TOPC:49; c in Cl R by A4,Th24; then c in Cl A by A6; hence c in A by A2,PRE_TOPC:52; end; Lm5: for S being compact Hausdorff TopLattice, N being net of S for c being Point of S, d being Element of S st c = d & (for x being Element of S holds x"/\" is continuous) & N is eventually-directed & c is_a_cluster_point_of N holds d is_>=_than rng the mapping of N proof let S be compact Hausdorff TopLattice, N be net of S, c be Point of S, d be Element of S such that A1: c = d and A2: for x being Element of S holds x"/\" is continuous and A3: N is eventually-directed and A4: c is_a_cluster_point_of N; let i be Element of S; assume i in rng the mapping of N; then consider iJ being set such that A5: iJ in dom the mapping of N and A6: (the mapping of N).iJ = i by FUNCT_1:def 5; reconsider i1 = iJ as Element of N by A5; A7: uparrow (N.i1) = {z where z is Element of S : z "/\" N.i1 = N.i1} by Lm1; A8: N is_eventually_in uparrow (N.i1) proof consider j being Element of N such that A9: for k being Element of N st j <= k holds N.i1 <= N.k by A3,WAYBEL_0:11; take j; let k be Element of N; assume j <= k; then N.i1 <= N.k by A9; hence N.k in uparrow (N.i1) by WAYBEL_0:18; end; uparrow N.i1 is closed by A2,Th27; then A10: Cl (uparrow N.i1) = uparrow N.i1 by PRE_TOPC:52; consider t being Element of N such that A11: for j being Element of N st t <= j holds N.j in uparrow N.i1 by A8,WAYBEL_0:def 11; reconsider A = N|t as subnet of N; A12: rng the mapping of A c= uparrow N.i1 proof let q be set; assume q in rng the mapping of A; then consider x being set such that A13: x in dom the mapping of A and A14: q = (the mapping of A).x by FUNCT_1:def 5; reconsider x as Element of A by A13; consider y being Element of N such that A15: y = x and A16: t <= y by Def7; A17: N.y in uparrow N.i1 by A11,A16; A.x = N.y by A15,Th16; hence q in uparrow N.i1 by A14,A17,WAYBEL_0:def 8; end; c is_a_cluster_point_of A proof let O be a_neighborhood of c; let i be Element of A; consider i2 being Element of N such that A18: i2 = i and A19: t <= i2 by Def7; N is_often_in O by A4,Def9; then consider j2 being Element of N such that A20: i2 <= j2 and A21: N.j2 in O by WAYBEL_0:def 12; t <= j2 by A19,A20,YELLOW_0:def 2; then j2 in the carrier of A by Def7; then reconsider j = j2 as Element of A; take j; A is full SubNetStr of N by Th14; hence i <= j by A18,A20,YELLOW_6:21; thus A.j in O by A21,Th16; end; then consider M being subnet of A such that A22: c in Lim M by Th32; consider f being map of M, A such that A23: the mapping of M = (the mapping of A)*f and for m being Element of A ex n being Element of M st for p being Element of M st n <= p holds m <= f.p by YELLOW_6:def 12; reconsider R = rng the mapping of M as Subset of S; R c= rng the mapping of A by A23,RELAT_1:45; then R c= uparrow N.i1 by A12,XBOOLE_1:1; then A24: Cl R c= Cl (uparrow N.i1) by PRE_TOPC:49; c in Cl R by A22,Th24; then c in uparrow (N.i1) by A10,A24; then consider z being Element of S such that A25: c = z & z "/\" N.i1 = N.i1 by A7; d >= N.i1 by A1,A25,YELLOW_0:25; hence d >= i by A6,WAYBEL_0:def 8; end; Lm6: for S being compact Hausdorff TopLattice, N being net of S for c being Point of S, d being Element of S st c = d & (for x being Element of S holds x"/\" is continuous) & c is_a_cluster_point_of N holds for b being Element of S st rng the mapping of N is_<=_than b holds d <= b proof let S be compact Hausdorff TopLattice, N be net of S, c be Point of S, d be Element of S such that A1: c = d and A2: for x being Element of S holds x"/\" is continuous and A3: c is_a_cluster_point_of N; let b be Element of S such that A4: rng the mapping of N is_<=_than b; A5: now let i be Element of N; A6: the carrier of N = dom the mapping of N by FUNCT_2:def 1; A7: N.i = (the mapping of N).i by WAYBEL_0:def 8; then reconsider Ni = N.i as Element of rng the mapping of N by A6,FUNCT_1:def 5; N.i in rng the mapping of N by A6,A7,FUNCT_1:def 5; then N.i <= b by A4,LATTICE3:def 9; then A8: b "/\" N.i = N.i by YELLOW_0:25; the carrier of S = [#]S by PRE_TOPC:12; then b in {b} & Ni in [#]S by TARSKI:def 1; hence N.i in {b} "/\" [#]S by A8,YELLOW_4:37; end; A9: {b} "/\" [#]S = {b "/\" y where y is Element of S: y in [#] S} by YELLOW_4:42; A10: rng the mapping of N c= {b} "/\" [#]S proof let y be set; assume y in rng the mapping of N; then consider x being set such that A11: x in dom the mapping of N and A12: y = (the mapping of N).x by FUNCT_1:def 5; reconsider x as Element of N by A11; y = N.x by A12,WAYBEL_0:def 8; hence y in {b} "/\" [#]S by A5; end; downarrow b is closed by A2,Th28; then {b} "/\" [#]S is closed by Th5; then c in {b} "/\" [#]S by A3,A10,Th34; then consider y being Element of S such that A13: c = b "/\" y and y in [#]S by A9; thus d <= b by A1,A13,YELLOW_0:23; end; theorem Th35: for S being compact Hausdorff TopLattice, c being Point of S for N being net of S st (for x being Element of S holds x"/\" is continuous) & N is eventually-directed & c is_a_cluster_point_of N holds c = sup N proof let S be compact Hausdorff TopLattice, c be Point of S, N be net of S such that A1: for x being Element of S holds x"/\" is continuous and A2: N is eventually-directed and A3: c is_a_cluster_point_of N; reconsider c' = c as Element of S; A4: c' is_>=_than rng the mapping of N by A1,A2,A3,Lm5; for b being Element of S st rng the mapping of N is_<=_than b holds c' <= b by A1,A3,Lm6; hence c = sup rng the mapping of N by A4,YELLOW_0:30 .= Sup the mapping of N by YELLOW_2:def 5 .= sup N by WAYBEL_2:def 1; end; Lm7: for S being compact Hausdorff TopLattice, N being net of S for c being Point of S, d being Element of S st c = d & (for x being Element of S holds x"/\" is continuous) & N is eventually-filtered & c is_a_cluster_point_of N holds d is_<=_than rng the mapping of N proof let S be compact Hausdorff TopLattice, N be net of S, c be Point of S, d be Element of S such that A1: c = d and A2: for x being Element of S holds x"/\" is continuous and A3: N is eventually-filtered and A4: c is_a_cluster_point_of N; let i be Element of S; assume i in rng the mapping of N; then consider iJ being set such that A5: iJ in dom the mapping of N and A6: (the mapping of N).iJ = i by FUNCT_1:def 5; reconsider i1 = iJ as Element of N by A5; A7: downarrow (N.i1) = {z where z is Element of S : z "\/" N.i1 = N.i1} by Lm2; A8: N is_eventually_in downarrow (N.i1) proof consider j being Element of N such that A9: for k being Element of N st j <= k holds N.i1 >= N.k by A3,WAYBEL_0:12; take j; let k be Element of N; assume j <= k; then N.i1 >= N.k by A9; hence N.k in downarrow (N.i1) by WAYBEL_0:17; end; downarrow N.i1 is closed by A2,Th28; then A10: Cl (downarrow N.i1) = downarrow N.i1 by PRE_TOPC:52; consider t being Element of N such that A11: for j being Element of N st t <= j holds N.j in downarrow N.i1 by A8,WAYBEL_0:def 11; reconsider A = N|t as subnet of N; A12: rng the mapping of A c= downarrow N.i1 proof let q be set; assume q in rng the mapping of A; then consider x being set such that A13: x in dom the mapping of A and A14: q = (the mapping of A).x by FUNCT_1:def 5; reconsider x as Element of A by A13; consider y being Element of N such that A15: y = x and A16: t <= y by Def7; A17: N.y in downarrow N.i1 by A11,A16; A.x = N.y by A15,Th16; hence q in downarrow N.i1 by A14,A17,WAYBEL_0:def 8; end; c is_a_cluster_point_of A proof let O be a_neighborhood of c; let i be Element of A; consider i2 being Element of N such that A18: i2 = i and A19: t <= i2 by Def7; N is_often_in O by A4,Def9; then consider j2 being Element of N such that A20: i2 <= j2 and A21: N.j2 in O by WAYBEL_0:def 12; t <= j2 by A19,A20,YELLOW_0:def 2; then j2 in the carrier of A by Def7; then reconsider j = j2 as Element of A; take j; A is full SubNetStr of N by Th14; hence i <= j by A18,A20,YELLOW_6:21; thus A.j in O by A21,Th16; end; then consider M being subnet of A such that A22: c in Lim M by Th32; consider f being map of M, A such that A23: the mapping of M = (the mapping of A)*f and for m being Element of A ex n being Element of M st for p being Element of M st n <= p holds m <= f.p by YELLOW_6:def 12; reconsider R = rng the mapping of M as Subset of S; R c= rng the mapping of A by A23,RELAT_1:45; then R c= downarrow N.i1 by A12,XBOOLE_1:1; then A24: Cl R c= Cl (downarrow N.i1) by PRE_TOPC:49; c in Cl R by A22,Th24; then c in downarrow (N.i1) by A10,A24; then consider z being Element of S such that A25: c = z & z "\/" N.i1 = N.i1 by A7; d <= N.i1 by A1,A25,YELLOW_0:24; hence d <= i by A6,WAYBEL_0:def 8; end; Lm8: for S being compact Hausdorff TopLattice, N being net of S for c being Point of S, d being Element of S st c = d & (for x being Element of S holds x"/\" is continuous) & c is_a_cluster_point_of N holds for b being Element of S st rng the mapping of N is_>=_than b holds d >= b proof let S be compact Hausdorff TopLattice, N be net of S, c be Point of S, d be Element of S such that A1: c = d and A2: for x being Element of S holds x"/\" is continuous and A3: c is_a_cluster_point_of N; let b be Element of S such that A4: rng the mapping of N is_>=_than b; A5: now let i be Element of N; A6: the carrier of N = dom the mapping of N by FUNCT_2:def 1; A7: N.i = (the mapping of N).i by WAYBEL_0:def 8; then reconsider Ni = N.i as Element of rng the mapping of N by A6,FUNCT_1:def 5; N.i in rng the mapping of N by A6,A7,FUNCT_1:def 5; then N.i >= b by A4,LATTICE3:def 8; then A8: b "\/" N.i = N.i by YELLOW_0:24; the carrier of S = [#]S by PRE_TOPC:12; then b in {b} & Ni in [#]S by TARSKI:def 1; hence N.i in {b} "\/" [#]S by A8,YELLOW_4:10; end; A9: {b} "\/" [#]S = {b "\/" y where y is Element of S: y in [#] S} by YELLOW_4:15; A10: rng the mapping of N c= {b} "\/" [#]S proof let y be set; assume y in rng the mapping of N; then consider x being set such that A11: x in dom the mapping of N and A12: y = (the mapping of N).x by FUNCT_1:def 5; reconsider x as Element of N by A11; y = N.x by A12,WAYBEL_0:def 8; hence y in {b} "\/" [#]S by A5; end; uparrow b is closed by A2,Th27; then {b} "\/" [#]S is closed by Th4; then c in {b} "\/" [#]S by A3,A10,Th34; then consider y being Element of S such that A13: c = b "\/" y and y in [#]S by A9; thus d >= b by A1,A13,YELLOW_0:22; end; theorem Th36: for S being compact Hausdorff TopLattice, c being Point of S for N being net of S st (for x being Element of S holds x"/\" is continuous) & N is eventually-filtered & c is_a_cluster_point_of N holds c = inf N proof let S be compact Hausdorff TopLattice, c be Point of S, N be net of S such that A1: for x being Element of S holds x"/\" is continuous and A2: N is eventually-filtered and A3: c is_a_cluster_point_of N; reconsider c' = c as Element of S; A4: c' is_<=_than rng the mapping of N by A1,A2,A3,Lm7; for b being Element of S st rng the mapping of N is_>=_than b holds c' >= b by A1,A3,Lm8; hence c = inf rng the mapping of N by A4,YELLOW_0:31 .= Inf the mapping of N by YELLOW_2:def 6 .= inf N by Def2; end; begin :: On The Topological Properties of Meet-Continuous Lattices :: Proposition 4.4 s. 32 (i) & (ii) => MC theorem Th37: for S being Hausdorff TopLattice st (for N being net of S st N is eventually-directed holds ex_sup_of N & sup N in Lim N) & (for x being Element of S holds x"/\" is continuous) holds S is meet-continuous proof let S be Hausdorff TopLattice such that A1: for N being net of S st N is eventually-directed holds ex_sup_of N & sup N in Lim N and A2: for x being Element of S holds x "/\" is continuous; for X being non empty directed Subset of S holds ex_sup_of X,S proof let X be non empty directed Subset of S; reconsider n = id X as Function of X, the carrier of S by FUNCT_2:9; set N = NetStr (#X,(the InternalRel of S)|_2 X,n#); N is eventually-directed by WAYBEL_2:20; then A3: ex_sup_of N by A1; rng the mapping of N = X by RELAT_1:71; hence ex_sup_of X,S by A3,Def3; end; hence S is up-complete by WAYBEL_0:75; for x being Element of S, M being net of S st M is eventually-directed holds x "/\" sup M = sup ({x} "/\" rng netmap (M,S)) proof let x be Element of S, M be net of S such that A4: M is eventually-directed; set xM = x "/\" M; A5: x "/\" M is eventually-directed by A4,WAYBEL_2:27; A6: x"/\" is continuous by A2; A7: sup M in Lim M by A1,A4; then reconsider M1 = M as convergent net of S by YELLOW_6:def 19; A8: sup xM in Lim xM by A1,A5; then reconsider xM as convergent net of S by YELLOW_6:def 19; A9: x "/\" lim M1 in Lim (x "/\" M1) by A6,Th26; thus x "/\" sup M = x "/\" lim M1 by A7,YELLOW_6:def 20 .= lim xM by A9,YELLOW_6:def 20 .= sup xM by A8,YELLOW_6:def 20 .= Sup the mapping of xM by WAYBEL_2:def 1 .= sup rng the mapping of xM by YELLOW_2:def 5 .= sup ({x} "/\" rng the mapping of M) by WAYBEL_2:23 .= sup ({x} "/\" rng netmap (M,S)) by WAYBEL_0:def 7; end; hence S is satisfying_MC by Th9; end; :: Proposition 4.4 s. 32 (ii) => (i) theorem Th38: for S being compact Hausdorff TopLattice st for x being Element of S holds x"/\" is continuous holds for N being net of S st N is eventually-directed holds ex_sup_of N & sup N in Lim N proof let S be compact Hausdorff TopLattice such that A1: for x being Element of S holds x "/\" is continuous; let N be net of S such that A2: N is eventually-directed; consider c being Point of S such that A3: c is_a_cluster_point_of N by Th30; A4: c = sup N by A1,A2,A3,Th35; thus ex_sup_of N proof set X = rng the mapping of N; reconsider d = c as Element of S; A5: X is_<=_than d by A1,A2,A3,Lm5; for b being Element of S st X is_<=_than b holds d <= b by A1,A3,Lm6; hence ex_sup_of rng the mapping of N, S by A5,YELLOW_0:15; end; for c, d being Point of S st c is_a_cluster_point_of N & d is_a_cluster_point_of N holds c = d proof let c, d be Point of S such that A6: c is_a_cluster_point_of N and A7: d is_a_cluster_point_of N; thus c = sup N by A1,A2,A6,Th35 .= d by A1,A2,A7,Th35; end; hence sup N in Lim N by A3,A4,Th33; end; :: Proposition 4.4 s. 32 (ii) => (i) dual theorem Th39: for S being compact Hausdorff TopLattice st for x being Element of S holds x"/\" is continuous holds for N being net of S st N is eventually-filtered holds ex_inf_of N & inf N in Lim N proof let S be compact Hausdorff TopLattice such that A1: for x being Element of S holds x "/\" is continuous; let N be net of S such that A2: N is eventually-filtered; consider c being Point of S such that A3: c is_a_cluster_point_of N by Th30; A4: c = inf N by A1,A2,A3,Th36; thus ex_inf_of N proof set X = rng the mapping of N; reconsider d = c as Element of S; A5: X is_>=_than d by A1,A2,A3,Lm7; for b being Element of S st X is_>=_than b holds d >= b by A1,A3,Lm8; hence ex_inf_of rng the mapping of N, S by A5,YELLOW_0:16; end; for c, d being Point of S st c is_a_cluster_point_of N & d is_a_cluster_point_of N holds c = d proof let c, d be Point of S such that A6: c is_a_cluster_point_of N and A7: d is_a_cluster_point_of N; thus c = inf N by A1,A2,A6,Th36 .= d by A1,A2,A7,Th36; end; hence inf N in Lim N by A3,A4,Th33; end; theorem for S being compact Hausdorff TopLattice st for x being Element of S holds x"/\" is continuous holds S is bounded proof let S be compact Hausdorff TopLattice such that A1: for x being Element of S holds x"/\" is continuous; thus S is lower-bounded proof reconsider x = inf (S opp+id) as Element of S; take x; A2: rng the mapping of S opp+id = rng id S by Def6 .= rng id the carrier of S by GRCAT_1:def 11 .= the carrier of S by RELAT_1:71; ex_inf_of S opp+id by A1,Th39; then A3: ex_inf_of the carrier of S, S by A2,Def4; x = Inf the mapping of S opp+id by Def2 .= "/\"(the carrier of S,S) by A2,YELLOW_2:def 6; hence x is_<=_than the carrier of S by A3,YELLOW_0:31; end; reconsider x = sup (S+id) as Element of S; take x; A4: rng the mapping of S+id = rng id S by Def5 .= rng id the carrier of S by GRCAT_1:def 11 .= the carrier of S by RELAT_1:71; ex_sup_of S+id by A1,Th38; then A5: ex_sup_of the carrier of S, S by A4,Def3; x = Sup the mapping of S+id by WAYBEL_2:def 1 .= "\/"(the carrier of S,S) by A4,YELLOW_2:def 5; hence x is_>=_than the carrier of S by A5,YELLOW_0:30; end; theorem for S being compact Hausdorff TopLattice st for x being Element of S holds x"/\" is continuous holds S is meet-continuous proof let S be compact Hausdorff TopLattice; assume A1: for x being Element of S holds x "/\" is continuous; then for N being net of S st N is eventually-directed holds ex_sup_of N & sup N in Lim N by Th38; hence S is meet-continuous by A1,Th37; end;

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