Copyright (c) 1997 Association of Mizar Users
environ vocabulary FINSET_1, SETFAM_1, TARSKI, BOOLE, CARD_1, SUBSET_1, RELAT_2, LATTICE3, ORDERS_1, WAYBEL_0, LATTICES, BHSP_3, ORDINAL2, WAYBEL_3, WAYBEL_6, RELAT_1, FILTER_0, WAYBEL_8, QUANTAL1, COMPTS_1, PRE_TOPC, YELLOW_1, WAYBEL_9, CANTOR_1, WAYBEL11, PROB_1, YELLOW_6, WAYBEL_2, FUNCT_1, TMAP_1, CONNSP_2, TOPS_1, YELLOW_8, TSP_1, YELLOW_0, WAYBEL_5, OPPCAT_1, WAYBEL14, RLVECT_3; notation TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, NAT_1, SETFAM_1, FINSET_1, DOMAIN_1, FUNCT_1, ORDERS_1, LATTICE3, CARD_1, STRUCT_0, PRE_TOPC, TOPS_1, TOPS_2, CONNSP_2, BORSUK_1, TMAP_1, CANTOR_1, TSP_1, COMPTS_1, YELLOW_0, YELLOW_1, YELLOW_3, YELLOW_4, YELLOW_6, YELLOW_7, YELLOW_8, WAYBEL_0, WAYBEL_1, WAYBEL_2, WAYBEL_3, WAYBEL_5, WAYBEL_6, WAYBEL_8, WAYBEL_9, WAYBEL11; constructors NAT_1, DOMAIN_1, TOPS_1, TOPS_2, TMAP_1, CANTOR_1, T_0TOPSP, YELLOW_4, YELLOW_8, WAYBEL_1, WAYBEL_3, WAYBEL_5, WAYBEL_6, WAYBEL_8, WAYBEL11, BORSUK_1, MEMBERED; clusters SUBSET_1, STRUCT_0, FINSET_1, LATTICE3, BORSUK_1, PRE_TOPC, CANTOR_1, YELLOW_1, YELLOW_4, YELLOW_6, YELLOW_8, WAYBEL_0, WAYBEL_3, WAYBEL_6, WAYBEL_7, WAYBEL_8, WAYBEL11, RELSET_1, MEMBERED, ZFMISC_1; requirements NUMERALS, BOOLE, SUBSET; definitions TARSKI, WAYBEL_0, WAYBEL_6, TMAP_1, YELLOW_8, COMPTS_1, WAYBEL_3, WAYBEL_1, WAYBEL_8, TOPS_2, XBOOLE_0; theorems TARSKI, ZFMISC_1, SUBSET_1, RELAT_1, DOMAIN_1, FUNCT_1, FUNCT_2, SETFAM_1, LATTICE3, CARD_2, ORDERS_1, PRE_TOPC, TOPS_1, TOPS_2, BORSUK_1, TMAP_1, CONNSP_2, COMPTS_1, CANTOR_1, YELLOW_0, YELLOW_1, YELLOW_2, YELLOW_3, YELLOW_4, YELLOW_6, YELLOW_7, YELLOW_8, WAYBEL_0, WAYBEL_2, WAYBEL_3, WAYBEL_4, WAYBEL_6, WAYBEL_8, WAYBEL11, WAYBEL12, XBOOLE_0, XBOOLE_1, XCMPLX_1; schemes NAT_1; begin :: Preliminaries theorem Th1: for X being set, F being finite Subset-Family of X ex G being finite Subset-Family of X st G c= F & union G = union F & for g being Subset of X st g in G holds not g c= union (G\{g}) proof let X be set; defpred P[Nat] means for F being finite Subset-Family of X st card F = $1 ex G being finite Subset-Family of X st G c= F & union G = union F & for g being Subset of X st g in G holds not g c= union (G\{g}); A1: P[0] proof let F be finite Subset-Family of X; assume A2: card F = 0; take G = F; thus G c= F; thus union G = union F; thus for g being Subset of X st g in G holds not g c= union (G\{g}) by A2, CARD_2:59; end; A3: now let n be Nat; assume A4: P[n]; thus P[n+1] proof let F be finite Subset-Family of X; assume A5: card F = n+1; per cases; suppose ex g being Subset of X st g in F & g c= union (F\{g}); then consider g being Subset of X such that A6: g in F & g c= union (F\{g}); reconsider FF = F\{g} as finite Subset-Family of X by SETFAM_1:def 7; {g} c= F by A6,ZFMISC_1:37; then A7: F = FF \/ {g} by XBOOLE_1:45; g in {g} by TARSKI:def 1; then not g in FF by XBOOLE_0:def 4; then card (FF \/ {g}) = card FF + 1 by CARD_2:54; then card FF = n by A5,A7,XCMPLX_1:2; then consider G being finite Subset-Family of X such that A8: G c= FF and A9: union G = union FF and A10: for g being Subset of X st g in G holds not g c= union (G\{g}) by A4; take G; A11: FF c= F by A7,XBOOLE_1:7; hence G c= F by A8,XBOOLE_1:1; A12: union FF c= union F by A11,ZFMISC_1:95; union F c= union FF proof let x be set; assume x in union F; then consider X being set such that A13: x in X & X in F by TARSKI:def 4; per cases by A7,A13,XBOOLE_0:def 2; suppose X in FF; then X c= union FF by ZFMISC_1:92; hence thesis by A13; suppose X in {g}; then X = g by TARSKI:def 1; hence thesis by A6,A13; end; hence union G = union F by A9,A12,XBOOLE_0:def 10; thus for g being Subset of X st g in G holds not g c= union (G\{g}) by A10; suppose A14: not ex g being Subset of X st g in F & g c= union (F\{g}); take G = F; thus G c= F; thus union G = union F; thus for g being Subset of X st g in G holds not g c= union (G\{g}) by A14; end; end; A15: for n being Nat holds P[n] from Ind(A1, A3); let F be finite Subset-Family of X; card F = card F; hence thesis by A15; end; Lm1: for S being 1-sorted, X, Y being Subset of S holds X c= Y` iff Y c= X` proof let S be 1-sorted, X, Y be Subset of S; Y = Y``; hence X c= Y` iff Y c= X` by PRE_TOPC:19; end; theorem Th2: for S being 1-sorted, X being Subset of S holds X` = the carrier of S iff X is empty proof let S be 1-sorted, X be Subset of S; hereby assume X` = the carrier of S; then X` = [#](the carrier of S) by SUBSET_1:def 4; then X = ([#](the carrier of S))`; then X = [#]S \ [#](the carrier of S) by PRE_TOPC:17 .= [#]S \ the carrier of S by SUBSET_1:def 4 .= {} by XBOOLE_1:37; hence X is empty; end; assume X is empty; then X = {}S; then X` = [#]S by PRE_TOPC:27; hence X` = the carrier of S by PRE_TOPC:12; end; theorem Th3: for R being antisymmetric with_infima transitive non empty RelStr, x, y being Element of R holds downarrow (x"/\"y) = (downarrow x) /\ downarrow y proof let R be antisymmetric with_infima transitive non empty RelStr, x,y be Element of R; now let z be set; hereby assume A1: z in downarrow (x"/\"y); then reconsider z' = z as Element of R; A2: z' <= (x"/\"y) by A1,WAYBEL_0:17; (x"/\"y) <= x & (x"/\"y) <= y by YELLOW_0:23; then z' <= x & z' <= y by A2,YELLOW_0:def 2; then z' in downarrow x & z' in downarrow y by WAYBEL_0:17; hence z in (downarrow x) /\ downarrow y by XBOOLE_0:def 3; end; assume A3: z in (downarrow x) /\ downarrow y; then A4: z in (downarrow x) & z in downarrow y by XBOOLE_0:def 3; reconsider z' = z as Element of R by A3; z' <= x & z' <= y by A4,WAYBEL_0:17; then x"/\"y >= z' by YELLOW_0:23; hence z in downarrow (x"/\"y) by WAYBEL_0:17; end; hence downarrow (x"/\"y) = (downarrow x) /\ downarrow y by TARSKI:2; end; theorem for R being antisymmetric with_suprema transitive non empty RelStr, x, y being Element of R holds uparrow (x"\/"y) = (uparrow x) /\ uparrow y proof let R be antisymmetric with_suprema transitive non empty RelStr, x,y be Element of R; now let z be set; hereby assume A1: z in uparrow (x"\/"y); then reconsider z' = z as Element of R; A2: z' >= (x"\/"y) by A1,WAYBEL_0:18; (x"\/"y) >= x & (x"\/"y) >= y by YELLOW_0:22; then z' >= x & z' >= y by A2,YELLOW_0:def 2; then z' in uparrow x & z' in uparrow y by WAYBEL_0:18; hence z in (uparrow x) /\ uparrow y by XBOOLE_0:def 3; end; assume A3: z in (uparrow x) /\ uparrow y; then A4: z in (uparrow x) & z in uparrow y by XBOOLE_0:def 3; reconsider z' = z as Element of R by A3; z' >= x & z' >= y by A4,WAYBEL_0:18; then x"\/"y <= z' by YELLOW_0:22; hence z in uparrow (x"\/"y) by WAYBEL_0:18; end; hence uparrow (x"\/"y) = (uparrow x) /\ uparrow y by TARSKI:2; end; theorem Th5: for L being complete antisymmetric (non empty RelStr), X being lower Subset of L st sup X in X holds X = downarrow sup X proof let L be complete antisymmetric (non empty RelStr), X be lower Subset of L such that A1: sup X in X; X is_<=_than sup X by YELLOW_0:32; hence X c= downarrow sup X by YELLOW_2:1; thus downarrow sup X c= X by A1,WAYBEL11:6; end; theorem for L being complete antisymmetric (non empty RelStr), X being upper Subset of L st inf X in X holds X = uparrow inf X proof let L be complete antisymmetric (non empty RelStr), X be upper Subset of L such that A1: inf X in X; X is_>=_than inf X by YELLOW_0:33; hence X c= uparrow inf X by YELLOW_2:2; thus uparrow inf X c= X by A1,WAYBEL11:42; end; theorem Th7: for R being non empty reflexive transitive RelStr, x, y being Element of R holds x << y iff uparrow y c= wayabove x proof let R be non empty reflexive transitive RelStr, x, y be Element of R; hereby assume A1: x << y; thus uparrow y c= wayabove x proof let z be set; assume A2: z in uparrow y; then reconsider z' = z as Element of R; y <= z' by A2,WAYBEL_0:18; then x << z' by A1,WAYBEL_3:2; hence z in wayabove x by WAYBEL_3:8; end; end; assume A3: uparrow y c= wayabove x; y <= y; then y in uparrow y by WAYBEL_0:18; hence x << y by A3,WAYBEL_3:8; end; theorem for R being non empty reflexive transitive RelStr, x, y being Element of R holds x << y iff downarrow x c= waybelow y proof let R be non empty reflexive transitive RelStr, x, y be Element of R; hereby assume A1: x << y; thus downarrow x c= waybelow y proof let z be set; assume A2: z in downarrow x; then reconsider z' = z as Element of R; z' <= x by A2,WAYBEL_0:17; then z' << y by A1,WAYBEL_3:2; hence z in waybelow y by WAYBEL_3:7; end; end; assume A3: downarrow x c= waybelow y; x <= x; then x in downarrow x by WAYBEL_0:17; hence x << y by A3,WAYBEL_3:7; end; theorem Th9: for R being complete reflexive antisymmetric (non empty RelStr), x being Element of R holds sup waybelow x <= x & x <= inf wayabove x proof let R be complete reflexive antisymmetric (non empty RelStr), x be Element of R; x is_>=_than waybelow x by WAYBEL_3:9; hence sup waybelow x <= x by YELLOW_0:32; x is_<=_than wayabove x by WAYBEL_3:10; hence thesis by YELLOW_0:33; end; theorem Th10: for L being lower-bounded antisymmetric non empty RelStr holds uparrow Bottom L = the carrier of L proof let L be lower-bounded antisymmetric non empty RelStr; set uL = uparrow Bottom L, cL = the carrier of L; now let x be set; thus x in uL implies x in cL; assume x in cL; then reconsider x' = x as Element of L; Bottom L <= x' by YELLOW_0:44; hence x in uL by WAYBEL_0:18; end; hence uparrow Bottom L = the carrier of L by TARSKI:2; end; theorem for L being upper-bounded antisymmetric non empty RelStr holds downarrow Top L = the carrier of L proof let L be upper-bounded antisymmetric non empty RelStr; set uL = downarrow Top L, cL = the carrier of L; now let x be set; thus x in uL implies x in cL; assume x in cL; then reconsider x' = x as Element of L; Top L >= x' by YELLOW_0:45; hence x in uL by WAYBEL_0:17; end; hence thesis by TARSKI:2; end; theorem Th12: for P being with_suprema Poset, x, y being Element of P holds (wayabove x)"\/"(wayabove y) c= uparrow (x"\/"y) proof let R be with_suprema Poset, x, y be Element of R; wayabove x c= uparrow x & wayabove y c= uparrow y by WAYBEL_3:11; then A1: (wayabove x)"\/"(wayabove y) c= (uparrow x)"\/"(uparrow y) by YELLOW_4:21; A2: uparrow x = uparrow {x} & uparrow y = uparrow {y} by WAYBEL_0:def 18; A3: {x}"\/"{y} = {x"\/"y} by YELLOW_4:19; A4: uparrow {x"\/"y} = uparrow (x"\/"y) by WAYBEL_0:def 18; (uparrow x)"\/"(uparrow y) c= uparrow ((uparrow x)"\/"(uparrow y)) by WAYBEL_0:16; then (uparrow x)"\/"(uparrow y) c= uparrow (x"\/"y) by A2,A3,A4,YELLOW_4:35; hence thesis by A1,XBOOLE_1:1; end; theorem for P being with_infima Poset, x, y being Element of P holds (waybelow x)"/\"(waybelow y) c= downarrow (x"/\"y) proof let R be with_infima Poset, x, y be Element of R; waybelow x c= downarrow x & waybelow y c= downarrow y by WAYBEL_3:11; then A1: (waybelow x)"/\"(waybelow y) c= (downarrow x)"/\"(downarrow y) by YELLOW_4:48; A2: downarrow x=downarrow {x} & downarrow y=downarrow {y} by WAYBEL_0:def 17; A3: {x}"/\"{y} = {x"/\"y} by YELLOW_4:46; A4: downarrow {x"/\"y} = downarrow (x"/\"y) by WAYBEL_0:def 17; (downarrow x)"/\"(downarrow y) c= downarrow ((downarrow x)"/\"(downarrow y)) by WAYBEL_0:16; then (downarrow x)"/\"(downarrow y) c= downarrow (x"/\"y) by A2,A3,A4,YELLOW_4:62; hence thesis by A1,XBOOLE_1:1; end; theorem Th14: for R being with_suprema non empty Poset, l being Element of R holds l is co-prime iff for x,y be Element of R st l <= x "\/" y holds l <= x or l <= y proof let R be with_suprema non empty Poset, l be Element of R; hereby assume l is co-prime; then A1: l~ is prime by WAYBEL_6:def 8; let x, y be Element of R; assume l <= x "\/" y; then A2: (x "\/" y)~ <= l~ by LATTICE3:9; (x "\/" y)~ = x"\/"y by LATTICE3:def 6 .= (x~)"/\"(y~) by YELLOW_7:23; then x~ <= l~ or y~ <= l~ by A1,A2,WAYBEL_6:def 6; hence l <= x or l <= y by LATTICE3:9; end; assume A3: for x,y be Element of R st l <= x "\/" y holds l <= x or l <= y; let x,y be Element of R~; assume A4: x "/\" y <= l~; ~(x "/\" y) = x "/\" y by LATTICE3:def 7 .= ~x"\/"~y by YELLOW_7:24; then l <= ~x"\/"~y by A4,YELLOW_7:2; then l <= ~x or l <= ~y by A3; hence x <= l~ or y <= l~ by YELLOW_7:2; end; theorem Th15: for P being complete (non empty Poset), V being non empty Subset of P holds downarrow inf V = meet {downarrow u where u is Element of P : u in V} proof let P be complete (non empty Poset), V be non empty Subset of P; set F = {downarrow u where u is Element of P : u in V}; consider u being set such that A1: u in V by XBOOLE_0:def 1; reconsider u as Element of P by A1; A2: downarrow u in F by A1; F c= bool the carrier of P proof let X be set; assume X in F; then consider u being Element of P such that A3: X = downarrow u & u in V; thus X in bool the carrier of P by A3; end; then reconsider F as Subset-Family of P by SETFAM_1:def 7; reconsider F as Subset-Family of P; now let x be set; hereby assume A4: x in downarrow inf V; then reconsider d = x as Element of P; A5: d <= inf V by A4,WAYBEL_0:17; now let Y be set; assume Y in F; then consider u being Element of P such that A6: Y = downarrow u & u in V; inf V is_<=_than V by YELLOW_0:33; then inf V <= u by A6,LATTICE3:def 8; then d <= u by A5,ORDERS_1:26; hence x in Y by A6,WAYBEL_0:17; end; hence x in meet F by A2,SETFAM_1:def 1; end; assume A7: x in meet F; then reconsider d = x as Element of P; now let b be Element of P; assume b in V; then downarrow b in F; then d in downarrow b by A7,SETFAM_1:def 1; hence d <= b by WAYBEL_0:17; end; then d is_<=_than V by LATTICE3:def 8; then d <= inf V by YELLOW_0:33; hence x in downarrow inf V by WAYBEL_0:17; end; hence thesis by TARSKI:2; end; theorem for P being complete (non empty Poset), V being non empty Subset of P holds uparrow sup V = meet {uparrow u where u is Element of P : u in V} proof let P be complete (non empty Poset), V be non empty Subset of P; set F = {uparrow u where u is Element of P : u in V}; consider u being set such that A1: u in V by XBOOLE_0:def 1; reconsider u as Element of P by A1; A2: uparrow u in F by A1; F c= bool the carrier of P proof let X be set; assume X in F; then consider u being Element of P such that A3: X = uparrow u & u in V; thus X in bool the carrier of P by A3; end; then reconsider F as Subset-Family of P by SETFAM_1:def 7; reconsider F as Subset-Family of P; now let x be set; hereby assume A4: x in uparrow sup V; then reconsider d = x as Element of P; A5: d >= sup V by A4,WAYBEL_0:18; now let Y be set; assume Y in F; then consider u being Element of P such that A6: Y = uparrow u & u in V; sup V is_>=_than V by YELLOW_0:32; then sup V >= u by A6,LATTICE3:def 9; then d >= u by A5,ORDERS_1:26; hence x in Y by A6,WAYBEL_0:18; end; hence x in meet F by A2,SETFAM_1:def 1; end; assume A7: x in meet F; then reconsider d = x as Element of P; now let b be Element of P; assume b in V; then uparrow b in F; then d in uparrow b by A7,SETFAM_1:def 1; hence d >= b by WAYBEL_0:18; end; then d is_>=_than V by LATTICE3:def 9; then d >= sup V by YELLOW_0:32; hence x in uparrow sup V by WAYBEL_0:18; end; hence thesis by TARSKI:2; end; definition let L be sup-Semilattice, x be Element of L; cluster compactbelow x -> directed; coherence proof set cX = compactbelow x; let xx, yy be Element of L such that A1: xx in cX & yy in cX; set z = xx "\/" yy; take z; xx <= x & yy <= x by A1,WAYBEL_8:4; then A2: z <= x by YELLOW_0:22; A3: xx <= z & yy <= z by YELLOW_0:22; xx is compact & yy is compact by A1,WAYBEL_8:4; then xx << xx & yy << yy by WAYBEL_3:def 2; then xx << z & yy << z by A3,WAYBEL_3:2; then z << z by WAYBEL_3:3; then z is compact by WAYBEL_3:def 2; hence z in cX by A2,WAYBEL_8:4; thus xx <= z & yy <= z by YELLOW_0:22; end; end; theorem Th17: :: See a parenthetical remark in the middle of p. 106. :: This fact is needed in the proof of II-1.11(ii), p. 105. for T being non empty TopSpace, S being irreducible Subset of T, V being Element of InclPoset the topology of T st V = S` holds V is prime proof let T be non empty TopSpace, S be irreducible Subset of T, V be Element of InclPoset the topology of T such that A1: V =S`; A2: the carrier of InclPoset the topology of T = the topology of T by YELLOW_1:1; A3: S is closed by YELLOW_8:def 4; set sL = the topology of T; let X, Y be Element of InclPoset sL; assume A4: X "/\" Y <= V; X in sL & Y in sL by A2; then reconsider X' = X, Y' = Y as Subset of T; X' is open & Y' is open by A2,PRE_TOPC:def 5; then X'` is closed & Y'` is closed by TOPS_1:30; then A5: (X'`)/\S is closed & (Y'`)/\S is closed by A3,TOPS_1:35; X' /\ Y' in sL by A2,PRE_TOPC:def 1; then X /\ Y = X "/\" Y by YELLOW_1:9; then X /\ Y c= V by A4,YELLOW_1:3; then S c= (X'/\Y')` by A1,Lm1; then S c= X'` \/ Y'` by WAYBEL12:6; then S = (X'` \/ Y'`)/\S by XBOOLE_1:28; then S = (X'`)/\S \/ (Y'`)/\S by XBOOLE_1:23; then S = (X'`)/\S or S = (Y'`)/\S by A5,YELLOW_8:def 4; then S c= X'` or S c= Y'` by XBOOLE_1:17; then X c= V or Y c= V by A1,Lm1; hence X <= V or Y <= V by YELLOW_1:3; end; theorem Th18: for T being non empty TopSpace, x, y be Element of InclPoset the topology of T holds x "\/" y = x \/ y & x "/\" y = x /\ y proof let T be non empty TopSpace, x, y be Element of InclPoset the topology of T; A1: the carrier of InclPoset the topology of T = the topology of T by YELLOW_1:1; then x in the topology of T & y in the topology of T; then reconsider x' = x, y' = y as Subset of T; x' is open & y' is open by A1,PRE_TOPC:def 5; then x' \/ y' is open by TOPS_1:37; then x' \/ y' in the topology of T by PRE_TOPC:def 5; hence x "\/" y = x \/ y by YELLOW_1:8; x' /\ y' in the topology of T by A1,PRE_TOPC:def 1; hence x "/\" y = x /\ y by YELLOW_1:9; end; theorem Th19: for T being non empty TopSpace, V being Element of InclPoset the topology of T holds V is prime iff for X, Y being Element of InclPoset the topology of T st X/\Y c= V holds X c= V or Y c= V proof let T be non empty TopSpace, V be Element of InclPoset the topology of T; hereby assume A1: V is prime; let X, Y be Element of InclPoset the topology of T; assume A2: X/\Y c= V; X/\Y = X"/\"Y by Th18; then X"/\"Y <= V by A2,YELLOW_1:3; then X <= V or Y <= V by A1,WAYBEL_6:def 6; hence X c= V or Y c= V by YELLOW_1:3; end; assume A3: for X, Y being Element of InclPoset the topology of T st X/\Y c= V holds X c= V or Y c= V; let X, Y be Element of InclPoset the topology of T such that A4: X "/\" Y <= V; X/\Y = X"/\"Y by Th18; then X/\Y c= V by A4,YELLOW_1:3; then X c= V or Y c= V by A3; hence X <= V or Y <= V by YELLOW_1:3; end; theorem for T being non empty TopSpace, V being Element of InclPoset the topology of T holds V is co-prime iff for X, Y being Element of InclPoset the topology of T st V c= X \/ Y holds V c= X or V c= Y proof let T be non empty TopSpace, V be Element of InclPoset the topology of T; hereby assume A1: V is co-prime; let X, Y be Element of InclPoset the topology of T; assume A2: V c= X \/ Y; X \/ Y = X "\/" Y by Th18; then V <= X"\/"Y by A2,YELLOW_1:3; then V <= X or V <= Y by A1,Th14; hence V c= X or V c= Y by YELLOW_1:3; end; assume A3: for X, Y being Element of InclPoset the topology of T st V c= X \/ Y holds V c= X or V c= Y; now let X, Y be Element of InclPoset the topology of T such that A4: V <= X"\/"Y; X \/ Y = X"\/"Y by Th18; then V c= X \/ Y by A4,YELLOW_1:3; then V c= X or V c= Y by A3; hence V <= X or V <= Y by YELLOW_1:3; end; hence thesis by Th14; end; definition let T be non empty TopSpace; cluster InclPoset the topology of T -> distributive; coherence proof let x, y, z be Element of InclPoset the topology of T; thus x "/\" (y "\/" z) = x /\ (y "\/" z) by Th18 .= x /\ (y \/ z) by Th18 .= x /\ y \/ x /\ z by XBOOLE_1:23 .= (x "/\" y) \/ x /\ z by Th18 .= (x "/\" y) \/ (x "/\" z) by Th18 .= (x "/\" y) "\/" (x "/\" z) by Th18; end; end; theorem Th21: for T being non empty TopSpace, L being TopLattice, t being Point of T, l being Point of L, X being Subset-Family of L st the TopStruct of T = the TopStruct of L & t = l & X is Basis of l holds X is Basis of t proof let T be non empty TopSpace, L be TopLattice, t be Point of T, l be Point of L, X be Subset-Family of L; assume A1: the TopStruct of T = the TopStruct of L; assume A2: t = l; assume A3: X c= the topology of L; assume A4: l in Intersect X; assume A5: for S being Subset of L st S is open & l in S ex V being Subset of L st V in X & V c= S; reconsider X' = X as Subset-Family of T by A1; now let S be Subset of T such that A6: S is open and A7: t in S; reconsider S' = S as Subset of L by A1; S in the topology of T by A6,PRE_TOPC:def 5; then S' is open by A1,PRE_TOPC:def 5; then consider V being Subset of L such that A8: V in X & V c= S' by A2,A5,A7; reconsider V as Subset of T by A1; take V; thus V in X' & V c= S by A8; end; hence X is Basis of t by A1,A2,A3,A4,YELLOW_8:def 2; end; theorem Th22: for L being TopLattice, x being Element of L st for X being Subset of L st X is open holds X is upper holds uparrow x is compact proof let L be TopLattice, x be Element of L such that A1: for X being Subset of L st X is open holds X is upper; set P = uparrow x; let F be Subset-Family of L such that A2: F is_a_cover_of P and A3: F is open; x <= x; then A4: x in P by WAYBEL_0:18; P c= union F by A2,COMPTS_1:def 1; then consider Y being set such that A5: x in Y & Y in F by A4,TARSKI:def 4; reconsider Y as Subset of L by A5; Y is open by A3,A5,TOPS_2:def 1; then Y is upper by A1; then A6: P c= Y by A5,WAYBEL11:42; reconsider G = {Y} as Subset-Family of L by SETFAM_1:def 7; reconsider G as Subset-Family of L; take G; thus G c= F by A5,ZFMISC_1:37; thus P c= union G by A6,ZFMISC_1:31; thus G is finite; end; begin :: Scott topology, continuation of WAYBEl11 reserve L for complete Scott TopLattice, x for Element of L, X, Y for Subset of L, V, W for Element of InclPoset sigma L, VV for Subset of InclPoset sigma L; definition let L be complete LATTICE; cluster sigma L -> non empty; coherence proof sigma L = the topology of ConvergenceSpace Scott-Convergence L by WAYBEL11:def 12; hence thesis; end; end; theorem Th23: sigma L = the topology of L proof the TopStruct of L = ConvergenceSpace Scott-Convergence L by WAYBEL11:32; hence sigma L = the topology of L by WAYBEL11:def 12; end; theorem Th24: X in sigma L iff X is open proof sigma L =the topology of L by Th23; hence thesis by PRE_TOPC:def 5;end; Lm2: for L being complete Scott TopLattice, V being filtered Subset of L, F being Subset-Family of L, CF being Subset of InclPoset sigma L st F = {downarrow u where u is Element of L : u in V} & CF = COMPLEMENT F holds CF is directed proof let L be complete Scott TopLattice, V be filtered Subset of L, F be Subset-Family of L, CF be Subset of InclPoset sigma L such that A1: F = {downarrow u where u is Element of L : u in V} and A2: CF = COMPLEMENT F; set IPs = InclPoset sigma L; A3: sigma L = the topology of L by Th23; then A4: the carrier of IPs = the topology of L by YELLOW_1:1; let x, y be Element of IPs such that A5: x in CF & y in CF; x in sigma L & y in sigma L by A3,A4; then reconsider x' = x, y' = y as Subset of L; x'` in F by A2,A5,YELLOW_8:13; then consider ux being Element of L such that A6: x'` = downarrow ux & ux in V by A1; y'` in F by A2,A5,YELLOW_8:13; then consider uy being Element of L such that A7: y'` = downarrow uy & uy in V by A1; consider uz being Element of L such that A8: uz in V & uz <= ux & uz <= uy by A6,A7,WAYBEL_0:def 2; (downarrow uz)` is open by WAYBEL11:12; then (downarrow uz)` in the topology of L by PRE_TOPC:def 5; then reconsider z = (downarrow uz)` as Element of IPs by A4 ; take z; downarrow uz in F by A1,A8; hence z in CF by A2,YELLOW_8:14; downarrow uz c= downarrow ux & downarrow uz c= downarrow uy by A8,WAYBEL_0:21; then (downarrow ux)` c= (downarrow uz)` & (downarrow uy)` c= (downarrow uz)` by PRE_TOPC:19; then x c= z & y c= z by A6,A7; hence x <= z & y <= z by YELLOW_1:3; end; theorem Th25: for X being filtered Subset of L st VV = {(downarrow x)` : x in X} holds VV is directed proof let V be filtered Subset of L; assume A1: VV = {(downarrow x)` : x in V}; set F = {downarrow u where u is Element of L : u in V}; F c= bool the carrier of L proof let x be set; assume x in F; then ex u being Element of L st x = downarrow u & u in V; hence thesis; end; then reconsider F as Subset-Family of L by SETFAM_1:def 7; reconsider F as Subset-Family of L; VV c= bool the carrier of L proof let x be set; assume x in VV; then ex u being Element of L st x = (downarrow u)` & u in V by A1; hence thesis; end; then reconsider VV as Subset-Family of L by SETFAM_1:def 7; reconsider VV as Subset-Family of L; now let x be set; hereby assume x in VV; then consider u being Element of L such that A2: x = (downarrow u)` & u in V by A1; downarrow u in F by A2; hence x in COMPLEMENT F by A2,YELLOW_8:14; end; assume A3: x in COMPLEMENT F; then reconsider X = x as Subset of L; X` in F by A3,YELLOW_8:13; then consider u being Element of L such that A4: X` = downarrow u & u in V; X = (downarrow u)` by A4; hence x in VV by A1,A4; end; then VV = COMPLEMENT F by TARSKI:2; hence thesis by Lm2; end; theorem Th26: X is open & x in X implies inf X << x proof assume that A1: X is open and A2: x in X; A3: X is upper property(S) by A1,WAYBEL11:15; now let D be non empty directed Subset of L; assume x <= sup D; then sup D in X by A2,A3,WAYBEL_0:def 20; then consider y being Element of L such that A4: y in D and A5: for x being Element of L st x in D & x >= y holds x in X by A3,WAYBEL11:def 3; take y; thus y in D by A4; A6: inf X is_<=_than X by YELLOW_0:33; y <= y; then y in X by A4,A5; hence inf X <= y by A6,LATTICE3:def 8; end; hence inf X << x by WAYBEL_3:def 1; end; :: p. 105 definition let R be non empty reflexive RelStr, f be map of [:R, R:], R; attr f is jointly_Scott-continuous means :Def1: for T being non empty TopSpace st the TopStruct of T = ConvergenceSpace Scott-Convergence R ex ft being map of [:T, T:], T st ft = f & ft is continuous; end; theorem Th27: :: Proposition 1.11 (i) p. 105 V = X implies (V is co-prime iff X is filtered upper) proof assume A1: V = X; A2: sigma L = the topology of ConvergenceSpace Scott-Convergence L by WAYBEL11:def 12; A3: the carrier of InclPoset sigma L = sigma L by YELLOW_1:1; then A4: X is upper by A1,A2,WAYBEL11:31; A5: the TopStruct of L = ConvergenceSpace Scott-Convergence L by WAYBEL11:32; A6: InclPoset the topology of L is with_suprema with_infima antisymmetric; hereby assume A7: V is co-prime; thus X is filtered proof let v, w be Element of L such that A8: v in X & w in X; (downarrow w)` is open & (downarrow v)` is open by WAYBEL11:12; then (downarrow w)` in sigma L & (downarrow v)` in sigma L by A2,A5,PRE_TOPC:def 5; then reconsider mdw = (downarrow w)`, mdv = (downarrow v)` as Element of InclPoset sigma L by A3; v <= v & w <= w; then v in downarrow v & w in downarrow w by WAYBEL_0:17; then not V c= (downarrow v)` & not V c= (downarrow w)` by A1,A8,YELLOW_6:10; then not V <= mdv & not V <= mdw by YELLOW_1:3; then not V <= mdv "\/" mdw by A2,A5,A6,A7,Th14; then A9: not V c= mdv "\/" mdw by YELLOW_1:3; mdv \/ mdw c= mdv "\/" mdw by A2,A5,A6,YELLOW_1:6; then A10: not V c= mdv \/ mdw by A9,XBOOLE_1:1; mdv \/ mdw = ((downarrow v) /\ downarrow w)` by WAYBEL12:6 .= (downarrow(v"/\"w))` by Th3; then X meets (downarrow(v"/\"w))`` by A1,A10,TOPS_1:20; then X/\(downarrow(v"/\"w))`` <> {} by XBOOLE_0:def 7; then X/\downarrow(v"/\"w) <> {}; then consider zz being set such that A11: zz in X/\downarrow(v"/\"w) by XBOOLE_0:def 1; A12: zz in X & zz in downarrow(v"/\"w) by A11,XBOOLE_0:def 3; reconsider zz as Element of L by A11; A13: zz <= v"/\"w by A12,WAYBEL_0:17; take z = v"/\"w; thus z in X by A4,A12,A13,WAYBEL_0:def 20; thus z <= v & z <= w by YELLOW_0:23; end; thus X is upper by A1,A2,A3,WAYBEL11:31; end; assume A14: X is filtered upper; assume not V is co-prime; then consider Xx, Y being Element of InclPoset sigma L such that A15: V <= Xx "\/" Y & not V <= Xx & not V <= Y by A2,A5,A6,Th14; Xx in sigma L & Y in sigma L by A3; then reconsider Xx' = Xx, Y' = Y as Subset of L; A16: Xx' is open & Y' is open by A2,A3,A5,PRE_TOPC:def 5; then Xx' \/ Y' is open by TOPS_1:37; then Xx \/ Y in sigma L by A2,A5,PRE_TOPC:def 5; then Xx \/ Y = Xx "\/" Y by YELLOW_1:8; then A17: V c= Xx \/ Y by A15,YELLOW_1:3; A18: not V c= Xx & not V c= Y by A15,YELLOW_1:3; then consider v being set such that A19: v in V & not v in Xx by TARSKI:def 3; consider w being set such that A20: w in V & not w in Y by A18,TARSKI:def 3; A21: Xx' is upper & Y' is upper by A16,WAYBEL11:def 4; reconsider v, w as Element of L by A1,A19,A20; A22: now assume A23: v"/\"w in Xx' \/ Y'; per cases by A23,XBOOLE_0:def 2; suppose A24: v"/\"w in Xx'; v"/\"w <= v by YELLOW_0:23; hence contradiction by A19,A21,A24,WAYBEL_0:def 20; suppose A25: v"/\"w in Y'; v"/\"w <= w by YELLOW_0:23; hence contradiction by A20,A21,A25,WAYBEL_0:def 20; end; v"/\"w in X by A1,A14,A19,A20,WAYBEL_0:41; hence contradiction by A1,A17,A22; end; theorem :: Proposition 1.11 (ii) p. 105 (V = X & ex x st X = (downarrow x)`) implies V is prime & V <> the carrier of L proof assume A1: V = X; A2: sigma L = the topology of ConvergenceSpace Scott-Convergence L by WAYBEL11:def 12; A3: the TopStruct of L = ConvergenceSpace Scott-Convergence L by WAYBEL11:32; given u being Element of L such that A4: X = (downarrow u)`; A5: Cl {u} = downarrow u by WAYBEL11:9; Cl {u} is irreducible by YELLOW_8:26; hence V is prime by A1,A2,A3,A4,A5,Th17; assume V = the carrier of L; hence contradiction by A1,A4,Th2; end; theorem Th29: :: Proposition 1.11 (iii) p. 105 V = X & sup_op L is jointly_Scott-continuous & V is prime & V <> the carrier of L implies ex x st X = (downarrow x)` proof assume that A1: V = X and A2: sup_op L is jointly_Scott-continuous and A3: V is prime and A4: V <> the carrier of L; A5: sigma L = the topology of ConvergenceSpace Scott-Convergence L by WAYBEL11:def 12; A6: the carrier of InclPoset sigma L = sigma L by YELLOW_1:1; A7: the TopStruct of L = ConvergenceSpace Scott-Convergence L by WAYBEL11:32; then A8: X is open by A1,A5,A6,PRE_TOPC:def 5; set A = X`; A is closed by A8,TOPS_1:30; then A9: A is directly_closed lower by WAYBEL11:7; take u = sup A; A10: A is directed proof given a, b being Element of L such that A11: a in A & b in A and A12: for z being Element of L holds not (z in A & a <= z & b <= z); a <= a"\/"b & b <= a"\/"b by YELLOW_0:22; then not a"\/"b in A by A12; then A13: a"\/"b in X by YELLOW_6:10; set LxL = [:L qua non empty TopSpace, L:]; consider Tsup being map of LxL, L such that A14: Tsup = sup_op L & Tsup is continuous by A2,A7,Def1; A15: Tsup"X is open by A8,A14,TOPS_2:55; A16: the carrier of LxL = [:the carrier of L, the carrier of L:] by BORSUK_1:def 5; then A17: [a,b] in the carrier of LxL by ZFMISC_1:def 2; Tsup. [a,b] = a"\/"b by A14,WAYBEL_2:def 5; then A18: [a,b] in Tsup"X by A13,A17,FUNCT_2:46; consider AA being Subset-Family of LxL such that A19:Tsup"X = union AA and A20:for e being set st e in AA ex X1 being Subset of L, Y1 being Subset of L st e = [:X1,Y1:] & X1 is open & Y1 is open by A15,BORSUK_1:45; consider AAe being set such that A21: [a,b] in AAe & AAe in AA by A18,A19,TARSKI:def 4; consider Va, Vb being Subset of L such that A22: AAe = [:Va, Vb:] & Va is open & Vb is open by A20,A21; reconsider Va' = Va, Vb' = Vb as Subset of L; now let x be set; hereby assume x in Tsup.:AAe; then consider cd being set such that A23: cd in the carrier of LxL and A24: cd in AAe & Tsup.cd = x by FUNCT_2:115; consider c, d being Element of L such that A25: cd = [c,d] by A16,A23,DOMAIN_1:9; reconsider c, d as Element of L; A26: x = c"\/"d by A14,A24,A25,WAYBEL_2:def 5; A27: c <= c"\/"d & d <= c"\/"d by YELLOW_0:22; A28: Va' is upper & Vb' is upper by A22,WAYBEL11:def 4; c in Va & d in Vb by A22,A24,A25,ZFMISC_1:106; then x in Va & x in Vb by A26,A27,A28,WAYBEL_0:def 20; hence x in Va/\Vb by XBOOLE_0:def 3; end; assume A29: x in Va/\Vb; then A30: x in Va & x in Vb by XBOOLE_0:def 3; reconsider c = x as Element of L by A29; A31: [c,c] in AAe by A22,A30,ZFMISC_1:106; A32: [c,c] in the carrier of LxL by A16,ZFMISC_1:106; c <= c; then c = c"\/"c by YELLOW_0:24; then c = Tsup. [c,c] by A14,WAYBEL_2:def 5; hence x in Tsup.:AAe by A31,A32,FUNCT_2:43; end; then A33: Tsup.:AAe = Va/\Vb by TARSKI:2; AAe c= union AA by A21,ZFMISC_1:92; then Tsup.:AAe c= Tsup.:(Tsup"X) & Tsup.:(Tsup"X) c= X by A19,FUNCT_1:145,RELAT_1:156; then A34: Tsup.:AAe c= X by XBOOLE_1:1; Va in sigma L & Vb in sigma L by A5,A7,A22,PRE_TOPC:def 5; then Va is Element of InclPoset the topology of L & Vb is Element of InclPoset the topology of L by A5,A6,A7; then A35: Va c= X or Vb c= X by A1,A3,A5,A7,A33,A34,Th19; a in Va & b in Vb by A21,A22,ZFMISC_1:106; hence contradiction by A11,A35,YELLOW_6:10; end; now assume A = {}; then A` = the carrier of L by Th2; hence contradiction by A1,A4; end; then u in A by A9,A10,WAYBEL11:def 2; then A = downarrow u by A9,Th5; hence X = (downarrow u)`; end; theorem Th30: :: Proposition 1.11 (iv) p. 105 L is continuous implies sup_op L is jointly_Scott-continuous proof assume A1: L is continuous; let T be non empty TopSpace such that A2: the TopStruct of T = ConvergenceSpace Scott-Convergence L; A3: the TopStruct of L = ConvergenceSpace Scott-Convergence L by WAYBEL11:32; A4: the carrier of T = the carrier of L by A2,YELLOW_6:def 27; set Tsup = sup_op L; A5: the carrier of [:T, T:] = [:the carrier of T, the carrier of T:] by BORSUK_1:def 5; then A6: the carrier of [:T,T:] = the carrier of [:L,L:] by A4,YELLOW_3:def 2; then reconsider Tsup as map of [:T, T:], T by A4; take Tsup; thus Tsup = sup_op L; for x being Point of [:T, T:] holds Tsup is_continuous_at x proof let ab be Point of [:T, T:]; consider a, b being Point of T such that A7: ab = [a,b] by A5,DOMAIN_1:9; let G be a_neighborhood of Tsup.ab; reconsider Tsab = Tsup.ab as Point of T; A8: Tsab in Int G by CONNSP_2:def 1; A9: Int G is open by TOPS_1:51; reconsider Tsab' = Tsab as Element of L by A4; reconsider basab = { wayabove q where q is Element of L: q << Tsab' } as Basis of Tsab' by A1,WAYBEL11:44; basab is Basis of Tsab by A2,A3,Th21; then consider V being Subset of T such that A10: V in basab & V c= Int G by A8,A9,YELLOW_8:def 2; consider u being Element of L such that A11: V = wayabove u & u << Tsab' by A10; reconsider a' = a, b' = b as Element of L by A4; reconsider Lc = L as continuous complete Scott TopLattice by A1; Lc = L; then A12: a' = sup waybelow a' & b' = sup waybelow b' by WAYBEL_3:def 5; A13: Tsab' = a'"\/"b' by A7,WAYBEL_2:def 5 .= sup ((waybelow a')"\/"(waybelow b')) by A12,WAYBEL_2:4; set D1 = waybelow a', D2 = waybelow b'; set D = D1"\/"D2; consider xy being Element of L such that A14: xy in D & u << xy by A1,A11,A13,WAYBEL_4:54; D = { x "\/" y where x, y is Element of L : x in D1 & y in D2 } by YELLOW_4:def 3; then consider x, y being Element of L such that A15: xy = x"\/"y & x in D1 & y in D2 by A14; A16: x << a' & y << b' by A15,WAYBEL_3:7; A17: wayabove x is open & wayabove y is open by A1,WAYBEL11:36; A18: a' in wayabove x & b' in wayabove y by A16,WAYBEL_3:8; reconsider wx = wayabove x, wy = wayabove y as Subset of T by A4; wayabove x in the topology of L & wayabove y in the topology of L by A17,PRE_TOPC:def 5; then A19: wx is open & wy is open by A2,A3,PRE_TOPC:def 5; reconsider H = [:wayabove x, wayabove y:] as Subset of [:T, T:] by A6; H is open by A19,BORSUK_1:46; then A20: H = Int H by TOPS_1:55; [a',b'] in H by A18,ZFMISC_1:106; then reconsider H as a_neighborhood of ab by A7,A20,CONNSP_2:def 1; take H; A21: Tsup.:H = (wayabove x)"\/"(wayabove y) by WAYBEL_2:35; A22: (wayabove x)"\/"(wayabove y) c= uparrow (x"\/"y) by Th12; uparrow (x"\/"y) c= wayabove u by A14,A15,Th7; then A23: Tsup.:H c= V by A11,A21,A22,XBOOLE_1:1; Int G c= G by TOPS_1:44; then V c= G by A10,XBOOLE_1:1; hence Tsup.:H c= G by A23,XBOOLE_1:1; end; hence Tsup is continuous by TMAP_1:49; end; theorem Th31: :: Corollary 1.12 p. 106 sup_op L is jointly_Scott-continuous implies L is sober proof assume A1: sup_op L is jointly_Scott-continuous; A2: sigma L = the topology of ConvergenceSpace Scott-Convergence L by WAYBEL11:def 12; A3: the carrier of InclPoset sigma L = sigma L by YELLOW_1:1; A4: the TopStruct of L = ConvergenceSpace Scott-Convergence L by WAYBEL11:32; let S be irreducible Subset of L; A5: S is non empty closed by YELLOW_8:def 4; then A6: S` is open by TOPS_1:29; A7: S` <> the carrier of L by Th2; S` in sigma L by A2,A4,A6,PRE_TOPC:def 5; then reconsider V = S` as Element of InclPoset sigma L by A3; V is prime by A2,A4,Th17; then consider p being Element of L such that A8: V = (downarrow p)` by A1,A7,Th29; A9: S = downarrow p by A8,TOPS_1:21; take p; A10: Cl {p} = downarrow p by WAYBEL11:9; hence p is_dense_point_of S by A9,YELLOW_8:27; let q be Point of L; assume q is_dense_point_of S; then A11: Cl {q} = S by A5,YELLOW_8:25; L is T_0 by WAYBEL11:10; hence p = q by A9,A10,A11,YELLOW_8:32; end; theorem :: Corollary 1.13 p. 106 L is continuous implies L is compact locally-compact sober Baire proof assume A1: L is continuous; A2: uparrow Bottom L = the carrier of L by Th10; A3: for X being Subset of L st X is open holds X is upper by WAYBEL11:def 4; then A4: uparrow Bottom L is compact by Th22; [#]L = the carrier of L by PRE_TOPC:12; hence L is compact by A2,A4,COMPTS_1:10; thus A5: L is locally-compact proof let x be Point of L, X be Subset of L such that A6: x in X and A7: X is open; reconsider x' = x as Element of L; set bas = { wayabove q where q is Element of L: q << x' }; A8: bas is Basis of x by A1,WAYBEL11:44; consider y being Element of L such that A9: y << x' & y in X by A1,A6,A7,WAYBEL11:43; A10: X is upper by A7,WAYBEL11:def 4; set Y = uparrow y; take Y; wayabove y in bas by A9; then A11: wayabove y is open & x in wayabove y by A8,YELLOW_8:21; wayabove y c= Y by WAYBEL_3:11; then wayabove y c= Int Y by A11,TOPS_1:56; hence x in Int Y by A11; thus Y c= X by A9,A10,WAYBEL11:42; thus Y is compact by A3,Th22; end; sup_op L is jointly_Scott-continuous by A1,Th30; hence L is sober by Th31; hence L is Baire by A5,WAYBEL12:48; end; theorem Th33: :: Theorem 1.14 (1) implies (2) p. 107 L is continuous & X in sigma L implies X = union {wayabove x : x in X} proof assume that A1: L is continuous and A2: X in sigma L; A3: X is open by A2,Th24; set WAV = {wayabove x where x is Element of L : x in X}; now let x be set; hereby assume A4: x in X; then reconsider x' = x as Element of L; set bas = { wayabove q where q is Element of L: q << x' }; A5: bas is Basis of x' by A1,WAYBEL11:44; consider q being Element of L such that A6: q << x' & q in X by A1,A3,A4,WAYBEL11:43; wayabove q in bas by A6; then A7: x' in wayabove q by A5,YELLOW_8:21; wayabove q in WAV by A6; hence x in union WAV by A7,TARSKI:def 4; end; assume x in union WAV; then consider Y being set such that A8: x in Y & Y in WAV by TARSKI:def 4; consider q being Element of L such that A9: Y = wayabove q & q in X by A8; X is upper by A3,WAYBEL11:def 4; then A10: uparrow q c= X by A9,WAYBEL11:42; wayabove q c= uparrow q by WAYBEL_3:11; then Y c= X by A9,A10,XBOOLE_1:1; hence x in X by A8; end; hence X = union {wayabove x where x is Element of L : x in X} by TARSKI:2; end; theorem :: Theorem 1.14 (2) implies (1) p. 107 (for X st X in sigma L holds X = union {wayabove x : x in X}) implies L is continuous proof assume A1: for X being Subset of L st X in sigma L holds X = union {wayabove x where x is Element of L : x in X}; thus for x being Element of L holds waybelow x is non empty directed; thus L is up-complete; let x be Element of L; set y = sup waybelow x, X = (downarrow y)`; A2: y <= x by Th9; assume A3: x <> sup waybelow x; now assume x in downarrow y; then x <= y by WAYBEL_0:17; hence contradiction by A2,A3,ORDERS_1:25; end; then A4: x in X by YELLOW_6:10; A5: X is open by WAYBEL11:12; set Z = {wayabove z where z is Element of L : z in X}; X in sigma L by A5,Th24; then X = union Z by A1; then consider Y being set such that A6: x in Y & Y in Z by A4,TARSKI:def 4; consider z being Element of L such that A7: Y = wayabove z & z in X by A6; z << x by A6,A7,WAYBEL_3:8; then A8: z in waybelow x by WAYBEL_3:7; y is_>=_than waybelow x by YELLOW_0:32; then z <= y by A8,LATTICE3:def 9; then z in downarrow y by WAYBEL_0:17; hence contradiction by A7,YELLOW_6:10; end; theorem :: Theorem 1.14 (1) implies (3 first conjunct) p. 107 L is continuous implies ex B being Basis of x st for X st X in B holds X is open filtered proof assume A1: L is continuous; then reconsider A = { wayabove q where q is Element of L: q << x } as Basis of x by WAYBEL11:44; set B = { V where V is Subset of L : ex q being Element of L st V c= wayabove q & q<<x & x in V & V is open filtered }; B c= bool the carrier of L proof let X be set; assume X in B; then ex V being Subset of L st X = V & ex q being Element of L st V c= wayabove q & q<<x & x in V & V is open filtered; hence thesis; end; then reconsider B as Subset-Family of L by SETFAM_1:def 7; reconsider B as Subset-Family of L; A2: B is Basis of x proof thus B c= the topology of L proof let Y be set; assume Y in B; then consider V being Subset of L such that A3: Y = V & ex q being Element of L st V c= wayabove q & q<<x & x in V & V is open filtered; thus thesis by A3,PRE_TOPC:def 5; end; thus x in Intersect B proof per cases; suppose B is empty; then Intersect B = the carrier of L by CANTOR_1:def 3; hence thesis; suppose A4: B is non empty; then A5: Intersect B = meet B by CANTOR_1:def 3; now let Y be set; assume Y in B; then consider V being Subset of L such that A6: Y = V & ex q being Element of L st V c= wayabove q & q<<x & x in V& V is open filtered; thus x in Y by A6; end; hence thesis by A4,A5,SETFAM_1:def 1; end; let S be Subset of L; assume S is open & x in S; then consider V being Subset of L such that A7: V in A & V c= S by YELLOW_8:def 2; consider q being Element of L such that A8: V = wayabove q & q << x by A7; consider F being Open Filter of L such that A9: x in F & F c= wayabove q by A1,A8,WAYBEL_6:8; take F; F is open by WAYBEL11:41; hence F in B by A8,A9; thus F c= S by A7,A8,A9,XBOOLE_1:1; end; now let Y be Subset of L; assume Y in B; then consider V being Subset of L such that A10: Y = V & ex q being Element of L st V c= wayabove q & q<<x & x in V & V is open filtered; thus Y is open filtered by A10; end; hence thesis by A2; end; theorem :: Theorem 1.14 (1) implies (3 second conjunct) p. 107 L is continuous implies InclPoset sigma L is continuous proof assume A1: L is continuous; set IPs = InclPoset the topology of L; A2: sigma L = the topology of L by Th23; A3: the carrier of IPs = the topology of L by YELLOW_1:1; A4: for x being Element of IPs holds waybelow x is non empty directed; IPs is satisfying_axiom_of_approximation proof let V be Element of IPs; set VV = {wayabove x where x is Element of L : x in V}; V in sigma L by A2,A3; then V is Subset of L; then A5: V = union VV by A1,A2,A3,Th33; set wV = waybelow V; now let x be set; hereby assume x in V; then consider xU being set such that A6: x in xU & xU in VV by A5,TARSKI:def 4; consider y being Element of L such that A7: xU = wayabove y & y in V by A6; wayabove y is open by A1,WAYBEL11:36; then xU in sigma L by A2,A7,PRE_TOPC:def 5; then reconsider xU as Element of IPs by A2,A3; xU << V proof let D be non empty directed Subset of IPs; assume V <= sup D; then V c= sup D by YELLOW_1:3; then V c= union D by YELLOW_1:22; then consider DD being set such that A8: y in DD & DD in D by A7,TARSKI:def 4; DD in sigma L by A2,A3,A8; then reconsider DD as Subset of L; DD is open by A3,A8,PRE_TOPC:def 5; then DD is upper by WAYBEL11:def 4; then A9: uparrow y c= DD by A8,WAYBEL11:42; wayabove y c= uparrow y by WAYBEL_3:11; then A10: wayabove y c= DD by A9,XBOOLE_1:1; reconsider d = DD as Element of IPs by A8; take d; thus d in D by A8; thus xU <= d by A7,A10,YELLOW_1:3; end; then xU in wV by WAYBEL_3:7; hence x in union wV by A6,TARSKI:def 4; end; assume x in union wV; then consider X being set such that A11: x in X & X in wV by TARSKI:def 4; reconsider X as Element of IPs by A11; X << V by A11,WAYBEL_3:7; then X <= V by WAYBEL_3:1; then X c= V by YELLOW_1:3; hence x in V by A11; end; then V = union waybelow V by TARSKI:2; hence V = sup waybelow V by YELLOW_1:22; end; hence thesis by A2,A4,WAYBEL_3:def 6; end; theorem Th37: :: Theorem 1.14 (3) implies (4) p. 107 (for x ex B being Basis of x st for Y st Y in B holds Y is open filtered) & InclPoset sigma L is continuous implies x = "\/" ({inf X : x in X & X in sigma L}, L) proof assume that A1: for x being Element of L ex B being Basis of x st for Y being Subset of L st Y in B holds Y is open filtered and A2: InclPoset sigma L is continuous; set IPs = InclPoset the topology of L; A3: sigma L = the topology of L by Th23; A4: the carrier of IPs = the topology of L by YELLOW_1:1; set IU = {inf V where V is Subset of L : x in V & V in sigma L}; set y = "\/"(IU,L); now let b be Element of L; assume b in IU; then consider V being Subset of L such that A5: b = inf V & x in V & V in sigma L; b is_<=_than V by A5,YELLOW_0:33; hence b <= x by A5,LATTICE3:def 8; end; then x is_>=_than IU by LATTICE3:def 9; then A6: y <= x by YELLOW_0:32; assume A7: x <> y; set VVl = (downarrow y)`; now assume x in downarrow y; then x <= y by WAYBEL_0:17; hence contradiction by A6,A7,ORDERS_1:25; end; then A8: x in VVl by YELLOW_6:10; VVl is open by WAYBEL11:12; then VVl in sigma L by Th24; then reconsider VVp = VVl as Element of IPs by A3,A4; IPs is satisfying_axiom_of_approximation by A2,A3,WAYBEL_3:def 6; then VVp = sup waybelow VVp by WAYBEL_3:def 5; then VVp = union waybelow VVp by YELLOW_1:22; then consider Vp being set such that A9: x in Vp & Vp in waybelow VVp by A8,TARSKI:def 4; reconsider Vp as Element of IPs by A9; A10: Vp << VVp by A9,WAYBEL_3:7; Vp in sigma L by A3,A4; then reconsider Vl = Vp as Subset of L; consider bas being Basis of x such that A11: for Y being Subset of L st Y in bas holds Y is open filtered by A1; Vl is open by A4,PRE_TOPC:def 5; then consider Ul being Subset of L such that A12: Ul in bas & Ul c= Vl by A9,YELLOW_8:def 2; A13: Ul is open & x in Ul by A12,YELLOW_8:21; then Ul in sigma L by A3,PRE_TOPC:def 5; then A14: inf Ul in IU by A13; y is_>=_than IU by YELLOW_0:32; then inf Ul <= y by A14,LATTICE3:def 9; then downarrow inf Ul c= downarrow y by WAYBEL_0:21; then A15: (downarrow y)` c= (downarrow inf Ul)` by PRE_TOPC:19; set F = {downarrow u where u is Element of L : u in Ul}; A16: downarrow x in F by A13; F c= bool the carrier of L proof let X be set; assume X in F; then consider u being Element of L such that A17: X = downarrow u & u in Ul; thus X in bool the carrier of L by A17; end; then F is non empty Subset-Family of L by A16,SETFAM_1:def 7 ; then reconsider F as non empty Subset-Family of L; downarrow inf Ul = meet F by A13,Th15; then A18: (downarrow inf Ul)` = union COMPLEMENT F by TOPS_2:12; A19: Ul is filtered by A11,A12; A20: (downarrow x)` in COMPLEMENT F by A16,YELLOW_8:14; COMPLEMENT F c= the topology of L proof let X be set; assume A21: X in COMPLEMENT F; then reconsider X' = X as Subset of L; X'` in F by A21,YELLOW_8:13; then consider u being Element of L such that A22: X'` = downarrow u & u in Ul; X' = (downarrow u)` by A22; then X' is open by WAYBEL11:12; hence X in the topology of L by PRE_TOPC:def 5; end; then reconsider CF = COMPLEMENT F as Subset of IPs by A4; A23: CF is directed by A3,A19,Lm2; VVp c= sup CF by A15,A18,YELLOW_1:22; then VVp <= sup CF by YELLOW_1:3; then consider d being Element of IPs such that A24: d in CF & Vp << d by A2,A3,A10,A20,A23,WAYBEL_4:54; d in sigma L by A3,A4; then reconsider d' = d as Subset of L; d'` in F by A24,YELLOW_8:13; then consider u being Element of L such that A25: d'` = downarrow u & u in Ul; Vp <= d by A24,WAYBEL_3:1; then A26: Vp c= d by YELLOW_1:3; u <= u; then u in downarrow u by WAYBEL_0:17; then not u in Vp by A25,A26,YELLOW_6:10; hence contradiction by A12,A25; end; theorem Th38: :: Theorem 1.14 (4) implies (1) p. 107 (for x holds x = "\/" ({inf X : x in X & X in sigma L}, L)) implies L is continuous proof assume A1: for x being Element of L holds x = "\/" ({inf V where V is Subset of L : x in V & V in sigma L}, L); thus for x being Element of L holds waybelow x is non empty directed; thus L is up-complete; let x be Element of L; set VV = {inf V where V is Subset of L : x in V & V in sigma L}; A2: VV c= waybelow x proof let d be set; assume d in VV; then consider V being Subset of L such that A3: inf V = d & x in V & V in sigma L; V is open by A3,Th24; then inf V << x by A3,Th26; hence d in waybelow x by A3,WAYBEL_3:7; end; ex_sup_of VV, L & ex_sup_of waybelow x, L by YELLOW_0:17; then A4: "\/" (VV, L) <= sup waybelow x by A2,YELLOW_0:34; x = "\/" (VV, L) & sup waybelow x <= x by A1,Th9; hence x = sup waybelow x by A4,ORDERS_1:25; end; theorem Th39: :: Theorem 1.14 (3) iff (5) p. 107 :: The conjunct InclPoset sigma L is continuous is dropped (for x ex B being Basis of x st for Y st Y in B holds Y is open filtered) iff (for V ex VV st V = sup VV & for W st W in VV holds W is co-prime) proof set IPs = InclPoset the topology of L; A1: sigma L = the topology of L by Th23; then A2: the carrier of IPs = sigma L by YELLOW_1:1; hereby assume A3: for x being Element of L ex X being Basis of x st for Y being Subset of L st Y in X holds Y is open filtered; let V be Element of InclPoset sigma L; V in sigma L by A1,A2; then reconsider Vl = V as Subset of L; A4: Vl is open by A1,A2,Th24; set X = {Y where Y is Subset of L : Y c= V & ex x being Element of L, bas being Basis of x st x in V & Y in bas & for Yx being Subset of L st Yx in bas holds Yx is open filtered}; now let YY be set; assume YY in X; then consider Y being Subset of L such that A5: Y = YY and Y c= V and A6: ex x being Element of L, bas being Basis of x st x in V & Y in bas & for Yx being Subset of L st Yx in bas holds Yx is open filtered; consider x be Element of L, bas being Basis of x such that A7: x in V & Y in bas by A6; Y is open by A7,YELLOW_8:21; then Y in sigma L by Th24; hence YY in the carrier of InclPoset sigma L by A5,YELLOW_1:1; end; then X c= the carrier of InclPoset sigma L by TARSKI:def 3; then reconsider X as Subset of InclPoset sigma L; take X; now let x be set; hereby assume A8: x in V; Vl = V; then reconsider d = x as Element of L by A8; consider bas being Basis of d such that A9: for Y being Subset of L st Y in bas holds Y is open filtered by A3; consider Y being Subset of L such that A10: Y in bas & Y c= V by A4,A8,YELLOW_8:22; x in Y & Y in X by A8,A9,A10,YELLOW_8:21; hence x in union X by TARSKI:def 4; end; assume x in union X; then consider YY being set such that A11: x in YY & YY in X by TARSKI:def 4; consider Y being Subset of L such that A12: Y = YY and A13: Y c= V and ex x being Element of L, bas being Basis of x st x in V & Y in bas & for Yx being Subset of L st Yx in bas holds Yx is open filtered by A11; thus x in V by A11,A12,A13; end; then V = union X by TARSKI:2; hence V = sup X by A1,YELLOW_1:22; let Yp be Element of InclPoset sigma L; assume Yp in X; then consider Y being Subset of L such that A14: Y = Yp and Y c= V and A15: ex x being Element of L, bas being Basis of x st x in V & Y in bas & for Yx being Subset of L st Yx in bas holds Yx is open filtered; consider x being Element of L, bas being Basis of x such that x in V and A16: Y in bas and A17: for Yx being Subset of L st Yx in bas holds Yx is open filtered by A15; A18: Y is open filtered by A16,A17; then Y is upper by WAYBEL11:def 4; hence Yp is co-prime by A14,A18,Th27; end; assume A19: for V being Element of InclPoset sigma L ex X being Subset of InclPoset sigma L st V = sup X & for x being Element of InclPoset sigma L st x in X holds x is co-prime; let x be Element of L; set bas = {V where V is Element of InclPoset sigma L : x in V & V is co-prime}; bas c= bool the carrier of L proof let VV be set; assume VV in bas; then ex V being Element of InclPoset sigma L st VV= V & x in V & V is co-prime; then VV in sigma L by A1,A2; hence thesis; end; then reconsider bas as Subset-Family of L by SETFAM_1:def 7; reconsider bas as Subset-Family of L; bas is Basis of x proof thus bas c= the topology of L proof let VV be set; assume VV in bas; then ex V being Element of InclPoset sigma L st VV= V & x in V & V is co-prime; hence thesis by A1,A2; end; now per cases; suppose bas is empty; then Intersect bas = the carrier of L by CANTOR_1:def 3; hence x in Intersect bas; suppose A20: bas is non empty; then A21: Intersect bas = meet bas by CANTOR_1:def 3; now let Y be set; assume Y in bas; then ex V being Element of InclPoset sigma L st Y = V & x in V & V is co-prime; hence x in Y; end; hence x in Intersect bas by A20,A21,SETFAM_1:def 1; end; hence x in Intersect bas; let S be Subset of L; assume A22: S is open & x in S; then S in sigma L by Th24; then reconsider S' = S as Element of IPs by A2; consider X being Subset of IPs such that A23: S' = sup X and A24: for x being Element of IPs st x in X holds x is co-prime by A1,A19; S' = union X by A23,YELLOW_1:22; then consider V being set such that A25: x in V & V in X by A22,TARSKI:def 4; V in sigma L by A2,A25; then reconsider V as Subset of L; reconsider Vp = V as Element of IPs by A25; take V; Vp is co-prime by A24,A25; hence V in bas by A1,A25; sup X is_>=_than X by YELLOW_0:32; then Vp <= sup X by A25,LATTICE3:def 9; hence V c= S by A23,YELLOW_1:3; end; then reconsider bas as Basis of x; take bas; let V be Subset of L; assume V in bas; then consider Vp being Element of InclPoset sigma L such that A26: V = Vp and x in Vp and A27: Vp is co-prime; thus V is open filtered by A1,A2,A26,A27,Th24,Th27; end; theorem :: Theorem 1.14 (5) iff (6) p. 107 (for V ex VV st V = sup VV & for W st W in VV holds W is co-prime) & InclPoset sigma L is continuous iff InclPoset sigma L is completely-distributive proof InclPoset sigma L = InclPoset the topology of L by Th23; hence thesis by WAYBEL_6:38; end; theorem :: Theorem 1.14 (6) iff (7) p. 107 InclPoset sigma L is completely-distributive iff InclPoset sigma L is continuous & (InclPoset sigma L) opp is continuous proof InclPoset sigma L = InclPoset the topology of L by Th23; hence thesis by WAYBEL_6:39; end; theorem :: Corollary 1.15 (1) implies (2) p. 108 L is algebraic implies ex B being Basis of L st B = {uparrow x : x in the carrier of CompactSublatt L} proof assume L is algebraic; then A1: L is satisfying_axiom_K by WAYBEL_8:def 4; set P = {uparrow k where k is Element of L : k in the carrier of CompactSublatt L}; P c= bool the carrier of L proof let x be set; assume x in P; then ex k being Element of L st x = uparrow k & k in the carrier of CompactSublatt L; hence thesis; end; then reconsider P as Subset-Family of L by SETFAM_1:def 7; reconsider P as Subset-Family of L; A2: P c= the topology of L proof let x be set; assume x in P; then consider k being Element of L such that A3: x = uparrow k & k in the carrier of CompactSublatt L; k is compact by A3,WAYBEL_8:def 1; then uparrow k is Open by WAYBEL_8:2; then uparrow k is open by WAYBEL11:41; hence x in the topology of L by A3,PRE_TOPC:def 5; end; now let x be Point of L; set B = {uparrow k where k is Element of L : uparrow k in P & x in uparrow k}; B c= bool the carrier of L proof let y be set; assume y in B; then ex k being Element of L st y = uparrow k & uparrow k in P & x in uparrow k; hence thesis; end; then reconsider B as Subset-Family of L by SETFAM_1:def 7; reconsider B as Subset-Family of L; B is Basis of x proof thus B c= the topology of L proof let y be set; assume y in B; then ex k being Element of L st y = uparrow k & uparrow k in P & x in uparrow k; hence thesis by A2; end; now per cases; suppose B is empty; then Intersect B = the carrier of L by CANTOR_1:def 3; hence x in Intersect B; suppose A4: B is non empty; then A5: Intersect B = meet B by CANTOR_1:def 3; now let Y be set; assume Y in B; then ex k being Element of L st Y = uparrow k & uparrow k in P & x in uparrow k; hence x in Y; end; hence x in Intersect B by A4,A5,SETFAM_1:def 1; end; hence x in Intersect B; let S be Subset of L such that A6: S is open and A7: x in S; reconsider x' = x as Element of L; A8: x = sup compactbelow x' by A1,WAYBEL_8:def 3; S is inaccessible by A6,WAYBEL11:def 4; then (compactbelow x') meets S by A7,A8,WAYBEL11:def 1; then consider k being set such that A9: k in compactbelow x' & k in S by XBOOLE_0:3; A10: compactbelow x' = downarrow x' /\ the carrier of CompactSublatt L by WAYBEL_8:5; reconsider k as Element of L by A9; take V = uparrow k; k in the carrier of CompactSublatt L by A9,A10,XBOOLE_0:def 3; then A11: uparrow k in P; k in downarrow x' by A9,A10,XBOOLE_0:def 3; then k <= x' by WAYBEL_0:17; then x in uparrow k by WAYBEL_0:18; hence V in B by A11; S is upper by A6,WAYBEL11:def 4; hence V c= S by A9,WAYBEL11:42; end; then reconsider B as Basis of x; take B; thus B c= P proof let y be set; assume y in B; then ex k being Element of L st y = uparrow k & uparrow k in P & x in uparrow k; hence y in P; end; end; then P is Basis of L by A2,YELLOW_8:23; hence thesis; end; theorem :: Corollary 1.15 (2) implies (3) p. 108 (ex B being Basis of L st B = {uparrow x :x in the carrier of CompactSublatt L}) implies InclPoset sigma L is algebraic & for V ex VV st V = sup VV & for W st W in VV holds W is co-prime proof given B being Basis of L such that A1: B = {uparrow k where k is Element of L : k in the carrier of CompactSublatt L}; set IPt = InclPoset the topology of L; set IPs = InclPoset sigma L; A2: sigma L = the topology of L by Th23; A3: the carrier of IPs = sigma L by YELLOW_1:1; A4: IPs = IPt by Th23; thus InclPoset sigma L is algebraic proof hereby let X be Element of IPs; reconsider X' = X as Element of IPt by Th23; compactbelow X' is non empty; hence compactbelow X is non empty by Th23; compactbelow X' is directed; hence compactbelow X is directed by A2; end; thus IPs is up-complete by A4; let X be Element of IPs; set cX = compactbelow X; A5: sup cX = union cX by A2,YELLOW_1:22; set GB = { G where G is Subset of L: G in B & G c= X }; X in sigma L by A3; then reconsider X' = X as Subset of L; X' is open by A3,Th24; then A6: X = union GB by YELLOW_8:18; now let x be set; hereby assume x in X; then consider GG being set such that A7: x in GG & GG in GB by A6,TARSKI:def 4; consider G being Subset of L such that A8: G = GG & G in B & G c= X by A7; consider k being Element of L such that A9: G = uparrow k & k in the carrier of CompactSublatt L by A1,A8; k is compact by A9,WAYBEL_8:def 1; then uparrow k is Open by WAYBEL_8:2; then uparrow k is open by WAYBEL11:41; then uparrow k in the topology of L by PRE_TOPC:def 5; then reconsider G as Element of IPs by A2,A3,A9; A10: G <= X by A8,YELLOW_1:3; for X being Subset of L st X is open holds X is upper by WAYBEL11:def 4; then uparrow k is compact by Th22; then G is compact by A2,A9,WAYBEL_3:36; then G in cX by A10,WAYBEL_8:4; hence x in union cX by A7,A8,TARSKI:def 4; end; assume x in union cX; then consider G being set such that A11: x in G & G in cX by TARSKI:def 4; reconsider G as Element of IPs by A11; G <= X by A11,WAYBEL_8:4; then G c= X by YELLOW_1:3; hence x in X by A11; end; hence X = sup compactbelow X by A5,TARSKI:2; end; let V be Element of InclPoset sigma L; set GB = { G where G is Subset of L: G in B & G c= V }; GB c= the carrier of IPs proof let x be set; assume x in GB; then consider G being Subset of L such that A12: G = x & G in B & G c= V; G is open by A12,YELLOW_8:19; hence x in the carrier of IPs by A3,A12,Th24; end; then reconsider GB as Subset of InclPoset sigma L; take GB; V in sigma L by A3; then reconsider V' = V as Subset of L; V' is open by A3,Th24; then V = union GB by YELLOW_8:18; hence V = sup GB by A2,YELLOW_1:22; let x be Element of InclPoset sigma L; assume x in GB; then consider G being Subset of L such that A13: G = x & G in B & G c= V; consider k being Element of L such that A14: G = uparrow k & k in the carrier of CompactSublatt L by A1,A13; thus x is co-prime by A13,A14,Th27; end; theorem :: Corollary 1.15 (3) implies (2) p. 108 :: The proof of ((3) implies (1)) is split into two parts :: This one proves ((3) implies (2)) and the next is ((2) implies (1)). InclPoset sigma L is algebraic & (for V ex VV st V = sup VV & for W st W in VV holds W is co-prime) implies ex B being Basis of L st B = {uparrow x : x in the carrier of CompactSublatt L} proof assume that A1: InclPoset sigma L is algebraic and A2: for V being Element of InclPoset sigma L ex X being Subset of InclPoset sigma L st V = sup X & for x being Element of InclPoset sigma L st x in X holds x is co-prime; set IPt = InclPoset the topology of L; set IPs = InclPoset sigma L; A3: sigma L = the topology of L by Th23; A4: the carrier of IPs = sigma L by YELLOW_1:1; A5: IPs = IPt by Th23; A6: InclPoset sigma L is satisfying_axiom_K by A1,WAYBEL_8:def 4; reconsider ips = InclPoset sigma L as algebraic LATTICE by A1,A5; A7: ips is continuous; (for x being Element of L ex X being Basis of x st for Y being Subset of L st Y in X holds Y is open filtered) by A2,Th39; then for x being Element of L holds x = "\/" ({inf V where V is Subset of L : x in V & V in sigma L}, L) by A7,Th37; then A8: L is continuous by Th38; set B = {uparrow k where k is Element of L : k in the carrier of CompactSublatt L}; B c= bool the carrier of L proof let x be set; assume x in B; then ex k being Element of L st x = uparrow k & k in the carrier of CompactSublatt L; hence thesis; end; then reconsider B as Subset-Family of L by SETFAM_1:def 7; reconsider B as Subset-Family of L; A9: B c= the topology of L proof let x be set; assume x in B; then consider k being Element of L such that A10: x = uparrow k & k in the carrier of CompactSublatt L; k is compact by A10,WAYBEL_8:def 1; then uparrow k is Open by WAYBEL_8:2; then uparrow k is open by WAYBEL11:41; hence x in the topology of L by A10,PRE_TOPC:def 5; end; now let x be Point of L; set Bx = {uparrow k where k is Element of L : uparrow k in B & x in uparrow k}; Bx c= bool the carrier of L proof let y be set; assume y in Bx; then ex k being Element of L st y = uparrow k & uparrow k in B & x in uparrow k; hence thesis; end; then reconsider Bx as Subset-Family of L by SETFAM_1:def 7; reconsider Bx as Subset-Family of L; Bx is Basis of x proof thus Bx c= the topology of L proof let y be set; assume y in Bx; then ex k being Element of L st y = uparrow k & uparrow k in B & x in uparrow k; hence thesis by A9; end; now per cases; suppose Bx is empty; then Intersect Bx = the carrier of L by CANTOR_1:def 3; hence x in Intersect Bx; suppose A11: Bx is non empty; then A12: Intersect Bx = meet Bx by CANTOR_1:def 3; now let Y be set; assume Y in Bx; then ex k being Element of L st Y = uparrow k & uparrow k in B & x in uparrow k; hence x in Y; end; hence x in Intersect Bx by A11,A12,SETFAM_1:def 1; end; hence x in Intersect Bx; let S be Subset of L such that A13: S is open and A14: x in S; S in the topology of L by A13,PRE_TOPC:def 5; then reconsider S' = S as Element of IPt by A3,A4; S' = sup compactbelow S' by A3,A6,WAYBEL_8:def 3; then S' = union compactbelow S' by YELLOW_1:22; then consider UA being set such that A15: x in UA & UA in compactbelow S' by A14,TARSKI:def 4; reconsider UA as Element of IPt by A15; A16: UA <= S' & UA is compact by A15,WAYBEL_8:4; consider F being Subset of InclPoset sigma L such that A17: UA = sup F and A18: for x being Element of InclPoset sigma L st x in F holds x is co-prime by A2,A3; A19: UA = union F by A3,A17,YELLOW_1:22; UA in the topology of L by A3,A4; then reconsider UA' = UA as Subset of L; A20: UA << UA by A16,WAYBEL_3:def 2; F is Subset of bool the carrier of L by A4,XBOOLE_1:1; then F is Subset-Family of L by SETFAM_1:def 7; then reconsider F' = F as Subset-Family of L; F' is open proof let P be Subset of L; assume P in F'; hence P is open by A3,A4,PRE_TOPC:def 5; end; then consider G being finite Subset of F' such that A21: UA c= union G by A19,A20,WAYBEL_3:34; union G c= union F by ZFMISC_1:95; then A22: UA = union G by A19,A21,XBOOLE_0:def 10; G is finite Subset of bool the carrier of L by XBOOLE_1:1; then G is finite Subset-Family of L by SETFAM_1:def 7; then reconsider G as finite Subset-Family of L; consider Gg being finite Subset-Family of L such that A23: Gg c= G and A24: union Gg = union G and A25: for g being Subset of L st g in Gg holds not g c= union (Gg\{g}) by Th1; consider U1 being set such that A26: x in U1 & U1 in Gg by A15,A21,A24,TARSKI:def 4; A27: Gg c= F by A23,XBOOLE_1:1; then U1 in F by A26; then reconsider U1 as Element of IPs; U1 in the topology of L by A3,A4; then reconsider U1' = U1 as Subset of L; U1 is co-prime by A18,A26,A27; then A28: U1' is filtered upper by Th27; set k = inf U1'; :: :: We simplify the proof from CCL a bit. :: :: Our proof follows their idea but we use UA (their U) as the compact set :: in InclPoset to get a contradiction after assumming that inf U1 is not :: in U1. :: now assume A29: not k in U1'; set D = {(downarrow u)` where u is Element of L : u in U1'}; consider u being set such that A30: u in U1' by A26; reconsider u as Element of L by A30; A31: (downarrow u)` in D by A30; D c= the topology of L proof let d be set; assume d in D; then consider u being Element of L such that A32: d = (downarrow u)` & u in U1'; (downarrow u)` is open by WAYBEL11:12; hence thesis by A32,PRE_TOPC:def 5; end; then reconsider D as non empty Subset of IPt by A3,A4,A31; A33: D is directed by A3,A28,Th25; now assume not UA c= union D; then consider l being set such that A34: l in UA' & not l in union D by TARSKI:def 3; reconsider l as Element of L by A34; consider Uk being set such that A35: l in Uk & Uk in Gg by A21,A24,A34,TARSKI:def 4; A36: Gg c= F by A23,XBOOLE_1:1; then Uk in F by A35; then reconsider Uk as Element of IPs; Uk in the topology of L by A3,A4; then reconsider Uk' = Uk as Subset of L; Uk is co-prime by A18,A35,A36; then A37: Uk' is filtered upper by Th27; A38: k is_<=_than U1' by YELLOW_0:33; now assume not l is_<=_than U1'; then consider m being Element of L such that A39: m in U1' & not l <= m by LATTICE3:def 8; (downarrow m)` in D by A39; then not l in (downarrow m)` by A34,TARSKI:def 4; then l in downarrow m by YELLOW_6:10; hence contradiction by A39,WAYBEL_0:17; end; then l <= k by YELLOW_0:33; then A40: k in Uk' by A35,A37,WAYBEL_0:def 20; A41: U1' c= Uk proof let u be set; assume A42: u in U1'; then reconsider d = u as Element of L; k <= d by A38,A42,LATTICE3:def 8; hence thesis by A37,A40,WAYBEL_0:def 20; end; U1' c= union (Gg\{U1'}) proof let u be set; assume A43: u in U1'; Uk in Gg\{U1'} by A29,A35,A40,ZFMISC_1:64; hence u in union (Gg\{U1'}) by A41,A43,TARSKI:def 4; end; hence contradiction by A25,A26; end; then UA c= sup D by YELLOW_1:22; then UA <= sup D by YELLOW_1:3; then consider d being Element of IPt such that A44: d in D and A45: UA <= d by A20,A33,WAYBEL_3:def 1; consider u being Element of L such that A46: d = (downarrow u)` & u in U1' by A44; U1 c= UA by A19,A26,A27,ZFMISC_1:92; then A47: u in UA by A46; UA c= d by A45,YELLOW_1:3; then A48: not u in downarrow u by A46,A47,YELLOW_6:10; u <= u; hence contradiction by A48,WAYBEL_0:17; end; then A49: uparrow k c= U1' by A28,WAYBEL11:42; A50: U1' c= uparrow k proof let x be set; assume A51: x in U1'; then reconsider x' = x as Element of L; k is_<=_than U1' by YELLOW_0:33; then k <= x' by A51,LATTICE3:def 8; hence x in uparrow k by WAYBEL_0:18; end; then A52: U1' = uparrow k by A49,XBOOLE_0:def 10; U1' is open by A3,A4,PRE_TOPC:def 5; then U1' is Open by A8,A28,WAYBEL11:46; then A53: k is compact by A52,WAYBEL_8:2; take V = uparrow k; k in the carrier of CompactSublatt L by A53,WAYBEL_8:def 1; then uparrow k in B; hence V in Bx by A26,A50; A54: UA c= S by A16,YELLOW_1:3; U1 c= UA by A22,A24,A26,ZFMISC_1:92; hence V c= S by A52,A54,XBOOLE_1:1; end; then reconsider Bx as Basis of x; take Bx; thus Bx c= B proof let y be set; assume y in Bx; then ex k being Element of L st y = uparrow k & uparrow k in B & x in uparrow k; hence y in B; end; end; then reconsider B as Basis of L by A9,YELLOW_8:23; take B; thus thesis; end; theorem :: Corollary 1.15 (2) implies (1) p. 108 (ex B being Basis of L st B = {uparrow x :x in the carrier of CompactSublatt L}) implies L is algebraic proof given B being Basis of L such that A1: B = {uparrow k where k is Element of L : k in the carrier of CompactSublatt L}; thus for x being Element of L holds compactbelow x is non empty directed; thus L is up-complete; let x be Element of L; set y = sup compactbelow x; set cx = compactbelow x; set dx = downarrow x; set dy = downarrow y; now assume A2: y <> x; for z be Element of L st z in dx holds z <= x by WAYBEL_0:17; then x is_>=_than dx by LATTICE3:def 9; then A3: sup dx <= x by YELLOW_0:32; A4: ex_sup_of cx, L & ex_sup_of dx, L by YELLOW_0:17; cx = dx /\ the carrier of CompactSublatt L by WAYBEL_8:5; then compactbelow x c= dx by XBOOLE_1:17; then sup compactbelow x <= sup dx by A4,YELLOW_0:34; then A5: y <= x by A3,ORDERS_1:26; now assume x in dy; then x <= y by WAYBEL_0:17; hence contradiction by A2,A5,ORDERS_1:25; end; then A6: x in dy` by YELLOW_6:10; set GB = { G where G is Subset of L: G in B & G c= dy`}; dy` is open by WAYBEL11:12; then dy` = union GB by YELLOW_8:18; then consider X being set such that A7: x in X & X in GB by A6,TARSKI:def 4; consider G being Subset of L such that A8: G = X & G in B & G c= dy` by A7; consider k being Element of L such that A9: G = uparrow k & k in the carrier of CompactSublatt L by A1,A8; A10: k <= x by A7,A8,A9,WAYBEL_0:18; k is compact by A9,WAYBEL_8:def 1; then A11: k in cx by A10,WAYBEL_8:4; y is_>=_than cx by YELLOW_0:32; then k <= y by A11,LATTICE3:def 9; then y in uparrow k by WAYBEL_0:18; then y <= y & not y in dy by A8,A9,YELLOW_6:10; hence contradiction by WAYBEL_0:17; end; hence x = sup compactbelow x; end;