Volume 15, 2003

University of Bialystok

Copyright (c) 2003 Association of Mizar Users

### The abstract of the Mizar article:

### On the Subcontinua of a Real Line

**by****Adam Grabowski**- Received June 12, 2003
- MML identifier: BORSUK_5

- [ Mizar article, MML identifier index ]

environ vocabulary BOOLE, ARYTM_1, PRE_TOPC, COMPTS_1, SEQ_1, RELAT_2, BORSUK_1, SUBSET_1, RCOMP_1, TOPMETR, CONNSP_1, ARYTM_3, SQUARE_1, RELAT_1, ORDINAL2, LATTICES, SEQ_2, INTEGRA1, FUNCT_1, TREAL_1, ARYTM, LIMFUNC1, BORSUK_2, PCOMPS_1, GRAPH_1, INCPROJ, CARD_1, SETFAM_1, RAT_1, METRIC_1, FINSET_1, ABSVALUE, IRRAT_1, POWER, INT_1, BORSUK_5, XREAL_0; notation TARSKI, XBOOLE_0, SUBSET_1, ORDINAL1, NUMBERS, XCMPLX_0, XREAL_0, REAL_1, ENUMSET1, SQUARE_1, RELAT_1, FUNCT_1, FUNCT_2, STRUCT_0, INT_1, PRE_TOPC, COMPTS_1, RCOMP_1, SEQ_4, TREAL_1, RCOMP_2, CONNSP_1, BORSUK_2, CARD_1, TOPMETR, PSCOMP_1, INTEGRA1, LIMFUNC1, IRRAT_1, FINSET_1, ABSVALUE, INCPROJ, RAT_1, METRIC_1, PCOMPS_1, NECKLACE; constructors TOPS_2, CONNSP_1, RCOMP_2, PSCOMP_1, INTEGRA1, BORSUK_3, COMPTS_1, YELLOW_8, PARTFUN1, LIMFUNC1, TREAL_1, POWER, PROB_1, BORSUK_2, ENUMSET1, NAT_1, INCPROJ, URYSOHN1, RAT_1, NECKLACE, MEMBERED; clusters XREAL_0, RELSET_1, TOPREAL6, STRUCT_0, PRE_TOPC, BORSUK_2, INTEGRA1, YELLOW13, TOPS_1, JORDAN6, BORSUK_4, FINSET_1, RAT_1, METRIC_1, INT_1, MEMBERED, ZFMISC_1; requirements NUMERALS, REAL, SUBSET, BOOLE, ARITHM; begin :: Preliminaries canceled; theorem :: BORSUK_5:2 for A, B, a being set st A c= B & B c= A \/ {a} holds A \/ {a} = B or A = B; theorem :: BORSUK_5:3 for x1, x2, x3, x4, x5, x6 being set holds { x1, x2, x3, x4, x5, x6 } = { x1, x3, x6 } \/ { x2, x4, x5 }; reserve x1, x2, x3, x4, x5, x6, x7 for set; definition let x1, x2, x3, x4, x5, x6 be set; pred x1, x2, x3, x4, x5, x6 are_mutually_different means :: BORSUK_5:def 1 x1 <> x2 & x1 <> x3 & x1 <> x4 & x1 <> x5 & x1 <> x6 & x2 <> x3 & x2 <> x4 & x2 <> x5 & x2 <> x6 & x3 <> x4 & x3 <> x5 & x3 <> x6 & x4 <> x5 & x4 <> x6 & x5 <> x6; end; definition let x1, x2, x3, x4, x5, x6, x7 be set; pred x1, x2, x3, x4, x5, x6, x7 are_mutually_different means :: BORSUK_5:def 2 x1 <> x2 & x1 <> x3 & x1 <> x4 & x1 <> x5 & x1 <> x6 & x1 <> x7 & x2 <> x3 & x2 <> x4 & x2 <> x5 & x2 <> x6 & x2 <> x7 & x3 <> x4 & x3 <> x5 & x3 <> x6 & x3 <> x7 & x4 <> x5 & x4 <> x6 & x4 <> x7 & x5 <> x6 & x5 <> x7 & x6 <> x7; end; theorem :: BORSUK_5:4 for x1, x2, x3, x4, x5, x6 being set st x1, x2, x3, x4, x5, x6 are_mutually_different holds card {x1, x2, x3, x4, x5, x6} = 6; theorem :: BORSUK_5:5 for x1, x2, x3, x4, x5, x6, x7 being set st x1, x2, x3, x4, x5, x6, x7 are_mutually_different holds card {x1, x2, x3, x4, x5, x6, x7} = 7; theorem :: BORSUK_5:6 { x1, x2, x3 } misses { x4, x5, x6 } implies x1 <> x4 & x1 <> x5 & x1 <> x6 & x2 <> x4 & x2 <> x5 & x2 <> x6 & x3 <> x4 & x3 <> x5 & x3 <> x6; theorem :: BORSUK_5:7 x1, x2, x3 are_mutually_different & x4, x5, x6 are_mutually_different & { x1, x2, x3 } misses { x4, x5, x6 } implies x1, x2, x3, x4, x5, x6 are_mutually_different; theorem :: BORSUK_5:8 x1, x2, x3, x4, x5, x6 are_mutually_different & { x1, x2, x3, x4, x5, x6 } misses { x7 } implies x1, x2, x3, x4, x5, x6, x7 are_mutually_different; theorem :: BORSUK_5:9 x1, x2, x3, x4, x5, x6, x7 are_mutually_different implies x7, x1, x2, x3, x4, x5, x6 are_mutually_different; theorem :: BORSUK_5:10 x1, x2, x3, x4, x5, x6, x7 are_mutually_different implies x1, x2, x5, x3, x6, x7, x4 are_mutually_different; theorem :: BORSUK_5:11 for T being non empty TopSpace, a, b being Point of T st (ex f being map of I[01], T st f is continuous & f.0 = a & f.1 = b) holds ex g being map of I[01], T st g is continuous & g.0 = b & g.1 = a; definition cluster R^1 -> arcwise_connected; end; definition cluster connected non empty TopSpace; end; begin :: Intervals theorem :: BORSUK_5:12 for A being Subset of REAL holds A is Subset of R^1; theorem :: BORSUK_5:13 [#]R^1 = REAL; definition let a be real number; redefine func left_closed_halfline a; :: LIMFUNC1 synonym ].-infty, a .]; redefine func left_open_halfline a; synonym ].-infty, a .[; redefine func right_closed_halfline a; synonym [. a,+infty.[; redefine func right_open_halfline a; synonym ]. a,+infty.[; end; theorem :: BORSUK_5:14 for a, b being real number holds a in ]. b,+infty.[ iff a > b; theorem :: BORSUK_5:15 for a, b being real number holds a in [. b,+infty.[ iff a >= b; theorem :: BORSUK_5:16 for a, b being real number holds a in ].-infty, b .] iff a <= b; theorem :: BORSUK_5:17 for a, b being real number holds a in ].-infty, b .[ iff a < b; theorem :: BORSUK_5:18 for a being real number holds REAL \ {a} = ].-infty, a .[ \/ ]. a,+infty.[; theorem :: BORSUK_5:19 for a, b, c, d being real number st a < b & b <= c holds [. a, b .] misses ]. c, d .]; theorem :: BORSUK_5:20 for a, b, c, d being real number st a < b & b <= c holds [. a, b .[ misses [. c, d .]; theorem :: BORSUK_5:21 for A, B being Subset of R^1, a, b, c, d being real number st a < b & b <= c & c < d & A = [. a, b .[ & B = ]. c, d .] holds A, B are_separated; theorem :: BORSUK_5:22 for a being real number holds REAL \ ].-infty, a .[ = [. a,+infty.[; theorem :: BORSUK_5:23 for a being real number holds REAL \ ].-infty, a .] = ]. a,+infty.[; theorem :: BORSUK_5:24 for a being real number holds REAL \ ]. a,+infty.[ = ].-infty, a .]; theorem :: BORSUK_5:25 for a being real number holds REAL \ [. a,+infty.[ = ].-infty, a .[; theorem :: BORSUK_5:26 for a being real number holds ].-infty, a .] misses ]. a,+infty.[; theorem :: BORSUK_5:27 for a being real number holds ].-infty, a .[ misses [. a,+infty.[; theorem :: BORSUK_5:28 for a, b, c being real number st a <= c & c <= b holds [.a,b.] \/ [.c,+infty.[ = [.a,+infty.[; theorem :: BORSUK_5:29 for a, b, c being real number st a <= c & c <= b holds ].-infty, c .] \/ [. a, b .] = ].-infty, b .]; theorem :: BORSUK_5:30 for T being 1-sorted, A being Subset of T holds { A } is Subset-Family of T; theorem :: BORSUK_5:31 for T being 1-sorted, A, B being Subset of T holds { A, B } is Subset-Family of T; theorem :: BORSUK_5:32 for T being 1-sorted, A, B, C being Subset of T holds { A, B, C } is Subset-Family of T; definition cluster -> real Element of RAT; end; definition cluster -> real Element of RealSpace; end; theorem :: BORSUK_5:33 for A being Subset of R^1, p being Point of RealSpace holds p in Cl A iff for r being real number st r > 0 holds Ball (p, r) meets A; theorem :: BORSUK_5:34 for p, q being Element of RealSpace st q >= p holds dist (p, q) = q - p; theorem :: BORSUK_5:35 for A being Subset of R^1 st A = RAT holds Cl A = the carrier of R^1; theorem :: BORSUK_5:36 for A being Subset of R^1, a, b being real number st A = ]. a, b .[ & a <> b holds Cl A = [. a, b .]; begin :: Rational and Irrational Numbers definition cluster number_e -> irrational; end; definition func IRRAT -> Subset of REAL equals :: BORSUK_5:def 3 REAL \ RAT; end; definition let a, b be real number; func RAT (a, b) -> Subset of REAL equals :: BORSUK_5:def 4 RAT /\ ]. a, b .[; func IRRAT (a, b) -> Subset of REAL equals :: BORSUK_5:def 5 IRRAT /\ ]. a, b .[; end; theorem :: BORSUK_5:37 for x being real number holds x is irrational iff x in IRRAT; definition cluster irrational (real number); end; definition cluster IRRAT -> non empty; end; theorem :: BORSUK_5:38 for a being rational number, b being irrational (real number) holds a + b is irrational; theorem :: BORSUK_5:39 for a being irrational (real number) holds -a is irrational; theorem :: BORSUK_5:40 for a being rational number, b being irrational (real number) holds a - b is irrational; theorem :: BORSUK_5:41 for a being rational number, b being irrational (real number) holds b - a is irrational; theorem :: BORSUK_5:42 for a being rational number, b being irrational (real number) st a <> 0 holds a * b is irrational; theorem :: BORSUK_5:43 for a being rational number, b being irrational (real number) st a <> 0 holds b / a is irrational; definition cluster irrational -> non zero (real number); end; theorem :: BORSUK_5:44 for a being rational number, b being irrational (real number) st a <> 0 holds a / b is irrational; theorem :: BORSUK_5:45 for r being irrational (real number) holds frac r is irrational; definition let r be irrational (real number); cluster frac r -> irrational; end; definition let a be irrational (real number); cluster -a -> irrational; end; definition let a be rational number, b be irrational (real number); cluster a + b -> irrational; cluster b + a -> irrational; cluster a - b -> irrational; cluster b - a -> irrational; end; definition cluster non zero (rational number); end; definition let a be non zero (rational number), b be irrational (real number); cluster a * b -> irrational; cluster b * a -> irrational; cluster a / b -> irrational; cluster b / a -> irrational; end; theorem :: BORSUK_5:46 for r being irrational (real number) holds 0 < frac r; theorem :: BORSUK_5:47 for a, b being real number st a < b ex p1, p2 being rational number st a < p1 & p1 < p2 & p2 < b; theorem :: BORSUK_5:48 for s1, s3, s4, l being real number holds s1 <= s3 & s1 < s4 & 0 < l & l < 1 implies s1 < (1-l)*s3+l*s4; theorem :: BORSUK_5:49 for s1, s3, s4, l being real number holds s3 < s1 & s4 <= s1 & 0 < l & l < 1 implies (1-l)*s3+l*s4 < s1; theorem :: BORSUK_5:50 for a, b being real number st a < b ex p being irrational (real number) st a < p & p < b; theorem :: BORSUK_5:51 for A being Subset of R^1 st A = IRRAT holds Cl A = the carrier of R^1; theorem :: BORSUK_5:52 for a, b, c being real number st a < b holds c in RAT (a,b) iff c is rational & a < c & c < b; theorem :: BORSUK_5:53 for a, b, c being real number st a < b holds c in IRRAT (a,b) iff c is irrational & a < c & c < b; theorem :: BORSUK_5:54 for A being Subset of R^1, a, b being real number st a < b & A = RAT (a, b) holds Cl A = [. a, b .]; theorem :: BORSUK_5:55 for A being Subset of R^1, a, b being real number st a < b & A = IRRAT (a, b) holds Cl A = [. a, b .]; theorem :: BORSUK_5:56 for T being connected TopSpace, A being closed open Subset of T holds A = {} or A = [#]T; theorem :: BORSUK_5:57 for A being Subset of R^1 st A is closed open holds A = {} or A = REAL; begin :: Topological Properties of Intervals theorem :: BORSUK_5:58 for A being Subset of R^1, a, b being real number st A = [. a, b .[ & a <> b holds Cl A = [. a, b .]; theorem :: BORSUK_5:59 for A being Subset of R^1, a, b being real number st A = ]. a, b .] & a <> b holds Cl A = [. a, b .]; theorem :: BORSUK_5:60 for A being Subset of R^1, a, b, c being real number st A = [. a, b .[ \/ ]. b, c .] & a < b & b < c holds Cl A = [. a, c .]; theorem :: BORSUK_5:61 for A being Subset of R^1, a being real number st A = {a} holds Cl A = {a}; theorem :: BORSUK_5:62 for A being Subset of REAL, B being Subset of R^1 st A = B holds A is open iff B is open; theorem :: BORSUK_5:63 for A being Subset of R^1, a being real number st A = ]. a,+infty.[ holds A is open; theorem :: BORSUK_5:64 for A being Subset of R^1, a being real number st A = ].-infty,a .[ holds A is open; theorem :: BORSUK_5:65 for A being Subset of R^1, a being real number st A = ].-infty,a.] holds A is closed; theorem :: BORSUK_5:66 for A being Subset of R^1, a being real number st A = [. a,+infty.[ holds A is closed; theorem :: BORSUK_5:67 for a being real number holds [. a,+infty.[ = {a} \/ ]. a,+infty.[; theorem :: BORSUK_5:68 for a being real number holds ].-infty,a.] = {a} \/ ].-infty,a.[; theorem :: BORSUK_5:69 for a being real number holds ]. a,+infty.[ c= [. a,+infty.[; theorem :: BORSUK_5:70 for a being real number holds ].-infty,a .[ c= ].-infty,a .]; definition let a be real number; cluster ]. a,+infty.[ -> non empty; cluster ].-infty,a .] -> non empty; cluster ].-infty,a .[ -> non empty; cluster [. a,+infty.[ -> non empty; end; theorem :: BORSUK_5:71 for a being real number holds ]. a,+infty.[ <> REAL; theorem :: BORSUK_5:72 for a being real number holds [. a,+infty.[ <> REAL; theorem :: BORSUK_5:73 for a being real number holds ].-infty, a .] <> REAL; theorem :: BORSUK_5:74 for a being real number holds ].-infty, a .[ <> REAL; theorem :: BORSUK_5:75 for A being Subset of R^1, a being real number st A = ]. a,+infty.[ holds Cl A = [. a,+infty.[; theorem :: BORSUK_5:76 for a being real number holds Cl ]. a,+infty.[ = [. a,+infty.[; theorem :: BORSUK_5:77 for A being Subset of R^1, a being real number st A = ].-infty, a .[ holds Cl A = ].-infty, a .]; theorem :: BORSUK_5:78 for a being real number holds Cl ].-infty, a .[ = ].-infty, a .]; theorem :: BORSUK_5:79 for A, B being Subset of R^1, b being real number st A = ].-infty, b .[ & B = ]. b,+infty.[ holds A, B are_separated; theorem :: BORSUK_5:80 for A being Subset of R^1, a, b being real number st a < b & A = [. a, b .[ \/ ]. b,+infty.[ holds Cl A = [. a,+infty.[; theorem :: BORSUK_5:81 for A being Subset of R^1, a, b being real number st a < b & A = ]. a, b .[ \/ ]. b,+infty.[ holds Cl A = [. a,+infty.[; theorem :: BORSUK_5:82 for A being Subset of R^1, a, b, c being real number st a < b & b < c & A = RAT (a,b) \/ ]. b, c .[ \/ ]. c,+infty.[ holds Cl A = [. a,+infty.[; theorem :: BORSUK_5:83 for A being Subset of R^1 holds A` = REAL \ A; theorem :: BORSUK_5:84 for a, b being real number st a < b holds IRRAT (a, b) misses RAT (a, b); theorem :: BORSUK_5:85 for a, b being real number st a < b holds REAL \ RAT (a, b) = ].-infty,a.] \/ IRRAT (a, b) \/ [.b,+infty.[; theorem :: BORSUK_5:86 for a, b, c being real number st a <= b & b < c holds not a in ]. b, c .[ \/ ]. c,+infty.[; theorem :: BORSUK_5:87 for a, b being real number st a < b holds not b in ]. a, b .[ \/ ]. b,+infty.[; theorem :: BORSUK_5:88 for a, b being real number st a < b holds [.a,+infty.[ \ (]. a, b .[ \/ ]. b,+infty.[) = {a} \/ {b}; theorem :: BORSUK_5:89 for A being Subset of R^1 st A = RAT (2,3) \/ ]. 3, 4 .[ \/ ]. 4,+infty.[ holds A` = ].-infty,2 .] \/ IRRAT(2,3) \/ {3} \/ {4}; theorem :: BORSUK_5:90 for A being Subset of R^1, a being real number st A = {a} holds A` = ].-infty, a .[ \/ ]. a,+infty.[; theorem :: BORSUK_5:91 for a, b being real number st a < b holds ]. a,+infty.[ /\ ].-infty, b .] = ]. a, b .]; theorem :: BORSUK_5:92 (].-infty, 1 .[ \/ ]. 1,+infty.[) /\ (].-infty, 2 .] \/ IRRAT(2,3) \/ {3} \/ {4}) = ].-infty, 1 .[ \/ ]. 1, 2 .] \/ IRRAT(2,3) \/ {3} \/ {4}; theorem :: BORSUK_5:93 for a, b being real number st a <= b holds ].-infty,b.[ \ {a} = ].-infty, a .[ \/ ]. a, b .[; theorem :: BORSUK_5:94 for a, b being real number st a <= b holds ]. a,+infty.[ \ {b} = ]. a, b .[ \/ ]. b,+infty.[; theorem :: BORSUK_5:95 for A being Subset of R^1, a, b being real number st a <= b & A = {a} \/ [. b,+infty.[ holds A` = ].-infty, a .[ \/ ]. a, b .[; theorem :: BORSUK_5:96 for A being Subset of R^1, a, b being real number st a < b & A = ].-infty, a .[ \/ ]. a, b .[ holds Cl A = ].-infty, b .]; theorem :: BORSUK_5:97 for A being Subset of R^1, a, b being real number st a < b & A = ].-infty, a .[ \/ ]. a, b .] holds Cl A = ].-infty, b .]; theorem :: BORSUK_5:98 for A being Subset of R^1, a being real number st A = ].-infty, a .] holds A` = ]. a ,+infty.[; theorem :: BORSUK_5:99 for A being Subset of R^1, a being real number st A = [. a ,+infty.[ holds A` = ].-infty, a .[; theorem :: BORSUK_5:100 for A being Subset of R^1, a, b, c being real number st a < b & b < c & A = ].-infty, a .[ \/ ]. a, b .] \/ IRRAT(b,c) \/ {c} holds Cl A = ].-infty, c .]; theorem :: BORSUK_5:101 for A being Subset of R^1, a, b, c, d being real number st a < b & b < c & A = ].-infty, a .[ \/ ]. a, b .] \/ IRRAT(b,c) \/ {c} \/ {d} holds Cl A = ].-infty, c .] \/ {d}; theorem :: BORSUK_5:102 for A being Subset of R^1, a, b being real number st a <= b & A = ].-infty, a .] \/ {b} holds A` = ]. a, b .[ \/ ]. b,+infty.[; theorem :: BORSUK_5:103 for a, b being real number holds [. a,+infty.[ \/ {b} <> REAL; theorem :: BORSUK_5:104 for a, b being real number holds ].-infty, a .] \/ {b} <> REAL; theorem :: BORSUK_5:105 for TS being TopStruct, A, B being Subset of TS st A <> B holds A` <> B`; theorem :: BORSUK_5:106 for A being Subset of R^1 st REAL = A` holds A = {}; begin :: Subcontinua of a real line definition cluster I[01] -> arcwise_connected; end; theorem :: BORSUK_5:107 for X being compact Subset of R^1, X' being Subset of REAL st X' = X holds X' is bounded_above bounded_below; theorem :: BORSUK_5:108 for X being compact Subset of R^1, X' being Subset of REAL, x being real number st x in X' & X' = X holds inf X' <= x & x <= sup X'; theorem :: BORSUK_5:109 for C being non empty compact connected Subset of R^1, C' being Subset of REAL st C = C' & [. inf C', sup C' .] c= C' holds [. inf C', sup C' .] = C'; theorem :: BORSUK_5:110 :: TOPREAL5:9 for A being connected Subset of R^1, a, b, c being real number st a <= b & b <= c & a in A & c in A holds b in A; theorem :: BORSUK_5:111 for A being connected Subset of R^1, a, b being real number st a in A & b in A holds [.a,b.] c= A; theorem :: BORSUK_5:112 for X being non empty compact connected Subset of R^1 holds X is non empty closed-interval Subset of REAL; theorem :: BORSUK_5:113 for A being non empty compact connected Subset of R^1 holds ex a, b being real number st a <= b & A = [. a, b .]; begin :: Sets with Proper Subsets Only definition let TS be TopStruct; let F be Subset-Family of TS; attr F is with_proper_subsets means :: BORSUK_5:def 6 not the carrier of TS in F; end; theorem :: BORSUK_5:114 for TS being TopStruct, F, G being Subset-Family of TS st F is with_proper_subsets & G c= F holds G is with_proper_subsets; definition let TS be non empty TopStruct; cluster with_proper_subsets Subset-Family of TS; end; theorem :: BORSUK_5:115 for TS being non empty TopStruct, A, B being with_proper_subsets Subset-Family of TS holds A \/ B is with_proper_subsets; definition let T be TopStruct, F be Subset-Family of T; attr F is open means :: BORSUK_5:def 7 :: TOPS_2:def 1 for P being Subset of T holds P in F implies P is open; attr F is closed means :: BORSUK_5:def 8 :: TOPS_2:def 2 for P being Subset of T holds P in F implies P is closed; end; definition let T be TopSpace; cluster open closed non empty Subset-Family of T; end;

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