Volume 15, 2003

University of Bialystok

Copyright (c) 2003 Association of Mizar Users

### The abstract of the Mizar article:

### On the Kuratowski Limit Operators

**by****Adam Grabowski**- Received August 12, 2003
- MML identifier: KURATO_2

- [ Mizar article, MML identifier index ]

environ vocabulary KURATO_2, WELLFND1, FRECHET2, REARRAN1, YELLOW_6, BOOLE, EUCLID, PRE_TOPC, COMPTS_1, SETFAM_1, METRIC_1, SUBSET_1, PCOMPS_1, RELAT_1, ORDINAL2, LATTICES, SEQ_2, PROB_1, SEQ_1, FUNCT_1, SEQM_3, FRECHET, ARYTM_3, CONNSP_2, ARYTM, TOPS_1, ARYTM_1, NEWTON, NORMSP_1, COMPLEX1, INT_1, FINSEQ_1, TOPREAL1, TOPREAL2, JORDAN9, JORDAN2C, SQUARE_1, GOBOARD9, ZF_REFLE, FUNCOP_1, MCART_1, WAYBEL_7; notation TARSKI, XBOOLE_0, SUBSET_1, ORDINAL1, NUMBERS, XCMPLX_0, ZFMISC_1, XREAL_0, REAL_1, NAT_1, SETFAM_1, MCART_1, STRUCT_0, PRE_TOPC, TOPS_1, COMPTS_1, RELAT_1, FUNCT_1, INT_1, FINSEQ_1, SEQ_1, TBSP_1, PARTFUN1, FUNCT_2, METRIC_1, PCOMPS_1, EUCLID, NORMSP_1, BORSUK_1, PROB_1, PSCOMP_1, LIMFUNC1, CONNSP_2, TOPREAL1, TOPREAL2, JORDAN2C, SEQM_3, GOBOARD9, FRECHET, FRECHET2, TOPRNS_1, JORDAN9, FUNCT_6, YELLOW_6; constructors REAL_1, JORDAN2C, GROUP_1, TOPREAL2, CONNSP_1, PSCOMP_1, TOPS_1, WEIERSTR, PROB_1, TOPRNS_1, FRECHET, NAT_1, JORDAN9, TBSP_1, CQC_SIM1, LIMFUNC1, GOBOARD9, YELLOW_6, FUNCT_6, FRECHET2; clusters XREAL_0, RELSET_1, NAT_1, INT_1, JORDAN1B, SUBSET_1, STRUCT_0, TOPS_1, BORSUK_1, EUCLID, FINSET_1, METRIC_1, TOPREAL6, GOBRD14, HAUSDORF, SEQ_1, SEQM_3, MEMBERED, FUNCT_2, PARTFUN1, ORDINAL2; requirements REAL, SUBSET, BOOLE, NUMERALS, ARITHM; begin :: Preliminaries theorem :: KURATO_2:1 for X, x being set, A being Subset of X st not x in A & x in X holds x in A`; theorem :: KURATO_2:2 for F being Function, i being set st i in dom F holds meet F c= F.i; theorem :: KURATO_2:3 for T being non empty 1-sorted, S1, S2 being SetSequence of the carrier of T holds S1 = S2 iff for n being Nat holds S1.n = S2.n; theorem :: KURATO_2:4 for A, B, C, D being set st A meets B & C meets D holds [: A, C :] meets [: B, D :]; definition let X be 1-sorted; cluster -> non empty SetSequence of the carrier of X; end; definition let T be non empty 1-sorted; cluster non-empty SetSequence of the carrier of T; end; definition let T be non empty 1-sorted; mode SetSequence of T is SetSequence of the carrier of T; canceled; end; scheme LambdaSSeq { X() -> non empty 1-sorted, F(set) -> Subset of X() } : ex f being SetSequence of X() st for n being Nat holds f.n = F(n); scheme ExTopStrSeq { R() -> non empty TopSpace, F(set) -> Subset of R() } : ex S be SetSequence of the carrier of R() st for n be Nat holds S.n = F(n); theorem :: KURATO_2:5 for X being non empty 1-sorted, F being SetSequence of the carrier of X holds rng F is Subset-Family of X; definition let X be non empty 1-sorted, F be SetSequence of the carrier of X; redefine func Union F -> Subset of X; redefine func meet F -> Subset of X; end; begin :: Lower and Upper Limit of Sequences of Subsets definition let X be non empty set; let S be Function of NAT, X; let k be Nat; func S ^\ k -> Function of NAT, X means :: KURATO_2:def 2 for n being Nat holds it.n = S.(n + k); end; definition let X be non empty 1-sorted, F be SetSequence of the carrier of X; func lim_inf F -> Subset of X means :: KURATO_2:def 3 ex f being SetSequence of X st it = Union f & for n being Nat holds f.n = meet (F ^\ n); func lim_sup F -> Subset of X means :: KURATO_2:def 4 ex f being SetSequence of X st it = meet f & for n being Nat holds f.n = Union (F ^\ n); end; theorem :: KURATO_2:6 for X being non empty 1-sorted, F being SetSequence of the carrier of X, x being set holds x in meet F iff for z being Nat holds x in F.z; theorem :: KURATO_2:7 for X being non empty 1-sorted, F being SetSequence of the carrier of X, x being set holds x in lim_inf F iff ex n being Nat st for k being Nat holds x in F.(n+k); theorem :: KURATO_2:8 for X being non empty 1-sorted, F being SetSequence of the carrier of X, x being set holds x in lim_sup F iff for n being Nat ex k being Nat st x in F.(n+k); theorem :: KURATO_2:9 for X being non empty 1-sorted, F being SetSequence of the carrier of X holds lim_inf F c= lim_sup F; theorem :: KURATO_2:10 for X being non empty 1-sorted, F being SetSequence of the carrier of X holds meet F c= lim_inf F; theorem :: KURATO_2:11 for X being non empty 1-sorted, F being SetSequence of the carrier of X holds lim_sup F c= Union F; theorem :: KURATO_2:12 for X being non empty 1-sorted, F being SetSequence of the carrier of X holds lim_inf F = (lim_sup Complement F)`; theorem :: KURATO_2:13 for X being non empty 1-sorted, A, B, C being SetSequence of the carrier of X st (for n being Nat holds C.n = A.n /\ B.n) holds lim_inf C = lim_inf A /\ lim_inf B; theorem :: KURATO_2:14 for X being non empty 1-sorted, A, B, C being SetSequence of the carrier of X st (for n being Nat holds C.n = A.n \/ B.n) holds lim_sup C = lim_sup A \/ lim_sup B; theorem :: KURATO_2:15 for X being non empty 1-sorted, A, B, C being SetSequence of the carrier of X st (for n being Nat holds C.n = A.n \/ B.n) holds lim_inf A \/ lim_inf B c= lim_inf C; theorem :: KURATO_2:16 for X being non empty 1-sorted, A, B, C being SetSequence of the carrier of X st (for n being Nat holds C.n = A.n /\ B.n) holds lim_sup C c= lim_sup A /\ lim_sup B; theorem :: KURATO_2:17 for X being non empty 1-sorted, A being SetSequence of the carrier of X, B being Subset of X st (for n being Nat holds A.n = B) holds lim_sup A = B; theorem :: KURATO_2:18 for X being non empty 1-sorted, A being SetSequence of the carrier of X, B being Subset of X st (for n being Nat holds A.n = B) holds lim_inf A = B; theorem :: KURATO_2:19 for X being non empty 1-sorted, A, B being SetSequence of the carrier of X, C being Subset of X st (for n being Nat holds B.n = C \+\ A.n) holds C \+\ lim_inf A c= lim_sup B; theorem :: KURATO_2:20 for X being non empty 1-sorted, A, B being SetSequence of the carrier of X, C being Subset of X st (for n being Nat holds B.n = C \+\ A.n) holds C \+\ lim_sup A c= lim_sup B; begin :: Ascending and Descending Families of Subsets definition let T be non empty 1-sorted, S be SetSequence of T; attr S is descending means :: KURATO_2:def 5 for i being Nat holds S.(i+1) c= S.i; attr S is ascending means :: KURATO_2:def 6 for i being Nat holds S.i c= S.(i+1); end; theorem :: KURATO_2:21 for f being Function st (for i being Nat holds f.(i+1) c= f.i) for i, j being Nat st i <= j holds f.j c= f.i; theorem :: KURATO_2:22 for T being non empty 1-sorted, C being SetSequence of T st C is descending holds for i, m being Nat st i >= m holds C.i c= C.m; theorem :: KURATO_2:23 for T being non empty 1-sorted, C being SetSequence of T st C is ascending holds for i, m being Nat st i >= m holds C.m c= C.i; theorem :: KURATO_2:24 for T being non empty 1-sorted, F being SetSequence of T, x being set st F is descending & ex k being Nat st for n being Nat st n > k holds x in F.n holds x in meet F; theorem :: KURATO_2:25 for T being non empty 1-sorted, F being SetSequence of T st F is descending holds lim_inf F = meet F; theorem :: KURATO_2:26 for T being non empty 1-sorted, F being SetSequence of T st F is ascending holds lim_sup F = Union F; begin :: Constant and Convergent Sequences definition let T be non empty 1-sorted, S be SetSequence of T; attr S is convergent means :: KURATO_2:def 7 lim_sup S = lim_inf S; end; theorem :: KURATO_2:27 for T being non empty 1-sorted, S being SetSequence of T st S is constant holds the_value_of S is Subset of T; definition let T be non empty 1-sorted, S be SetSequence of T; redefine attr S is constant means :: KURATO_2:def 8 ex A being Subset of T st for n being Nat holds S.n = A; end; definition let T be non empty 1-sorted; cluster constant -> convergent ascending descending SetSequence of T; end; definition let T be non empty 1-sorted; cluster constant non empty SetSequence of T; end; definition let T be non empty 1-sorted, S be convergent SetSequence of T; func Lim_K S -> Subset of T means :: KURATO_2:def 9 it = lim_sup S & it = lim_inf S; end; theorem :: KURATO_2:28 for X being non empty 1-sorted, F being convergent SetSequence of X, x being set holds x in Lim_K F iff ex n being Nat st for k being Nat holds x in F.(n+k); begin :: Topological Lemmas reserve n for Nat; definition let f be FinSequence of the carrier of TOP-REAL 2; cluster L~f -> closed; end; theorem :: KURATO_2:29 for r being real number, M being non empty Reflexive MetrStruct, x being Element of M st 0 < r holds x in Ball (x,r); theorem :: KURATO_2:30 for x being Point of Euclid n, r being real number holds Ball (x, r) is open Subset of TOP-REAL n; theorem :: KURATO_2:31 for p, q being Point of TOP-REAL n, p', q' being Point of Euclid n st p = p' & q = q' holds dist (p', q') = |. p - q .|; theorem :: KURATO_2:32 for p being Point of Euclid n, x, p' being Point of TOP-REAL n, r being real number st p = p' & x in Ball (p, r) holds |. x - p' .| < r; theorem :: KURATO_2:33 for p being Point of Euclid n, x, p' being Point of TOP-REAL n, r being real number st p = p' & |. x - p' .| < r holds x in Ball (p, r); theorem :: KURATO_2:34 for n being Nat, r being Point of TOP-REAL n, X being Subset of TOP-REAL n st r in Cl X holds ex seq being Real_Sequence of n st rng seq c= X & seq is convergent & lim seq = r; definition let M be non empty MetrSpace; cluster TopSpaceMetr M -> first-countable; end; definition let n be Nat; cluster TOP-REAL n -> first-countable; end; theorem :: KURATO_2:35 for p being Point of Euclid n, q being Point of TOP-REAL n, r being real number st p = q & r > 0 holds Ball (p, r) is a_neighborhood of q; theorem :: KURATO_2:36 for A being Subset of TOP-REAL n, p being Point of TOP-REAL n, p' being Point of Euclid n st p = p' holds p in Cl A iff for r being real number st r > 0 holds Ball (p', r) meets A; theorem :: KURATO_2:37 for x, y being Point of TOP-REAL n, x' being Point of Euclid n st x' = x & x <> y ex r being Real st not y in Ball (x', r); theorem :: KURATO_2:38 for S being Subset of TOP-REAL n holds S is non Bounded iff for r being Real st r > 0 holds ex x, y being Point of Euclid n st x in S & y in S & dist (x, y) > r; theorem :: KURATO_2:39 for a, b being real number, x, y being Point of Euclid n st Ball (x, a) meets Ball (y, b) holds dist (x, y) < a + b; theorem :: KURATO_2:40 for a, b, c being real number, x, y, z being Point of Euclid n st Ball (x, a) meets Ball (z, c) & Ball (z, c) meets Ball (y, b) holds dist (x, y) < a + b + 2*c; theorem :: KURATO_2:41 for X, Y being non empty TopSpace, x being Point of X, y being Point of Y, V being Subset of [: X, Y :] holds V is a_neighborhood of [: {x}, {y} :] iff V is a_neighborhood of [ x, y ]; scheme TSubsetEx { S() -> non empty TopStruct, P[set] } : ex X being Subset of S() st for x being Point of S() holds x in X iff P[x]; scheme TSubsetUniq { S() -> TopStruct, P[set] } : for A1, A2 being Subset of S() st (for x being Point of S() holds x in A1 iff P[x]) & (for x being Point of S() holds x in A2 iff P[x]) holds A1 = A2; definition let T be non empty TopStruct; let S be SetSequence of the carrier of T; let i be Nat; redefine func S.i -> Subset of T; end; theorem :: KURATO_2:42 for T be non empty 1-sorted, S being SetSequence of the carrier of T, R being Seq_of_Nat holds S * R is SetSequence of T; theorem :: KURATO_2:43 id NAT is increasing Seq_of_Nat; definition cluster id NAT -> real-yielding; end; begin :: Subsequences definition let T be non empty 1-sorted, S be SetSequence of the carrier of T; mode subsequence of S -> SetSequence of T means :: KURATO_2:def 10 ex NS being increasing Seq_of_Nat st it = S * NS; end; theorem :: KURATO_2:44 for T being non empty 1-sorted, S being SetSequence of the carrier of T holds S is subsequence of S; theorem :: KURATO_2:45 for T being non empty 1-sorted, S being SetSequence of T, S1 being subsequence of S holds rng S1 c= rng S; theorem :: KURATO_2:46 for T being non empty 1-sorted, S1 being SetSequence of the carrier of T, S2 being subsequence of S1, S3 being subsequence of S2 holds S3 is subsequence of S1; theorem :: KURATO_2:47 for T being non empty 1-sorted, F, G being SetSequence of the carrier of T, A being Subset of T st G is subsequence of F & for i being Nat holds F.i = A holds G = F; theorem :: KURATO_2:48 for T being non empty 1-sorted, A being constant SetSequence of T, B being subsequence of A holds A = B; theorem :: KURATO_2:49 for T being non empty 1-sorted, S being SetSequence of the carrier of T, R being subsequence of S, n being Nat holds ex m being Nat st m >= n & R.n = S.m; definition let T be non empty 1-sorted, X be constant SetSequence of T; cluster -> constant subsequence of X; end; scheme SubSeqChoice { T() -> non empty TopSpace, S() -> SetSequence of the carrier of T(), P[set]} : ex S1 being subsequence of S() st for n being Nat holds P[S1.n] provided for n being Nat ex m being Nat st n <= m & P[S().m]; begin :: Lower Topological Limit definition let T be non empty TopSpace; let S be SetSequence of the carrier of T; func Lim_inf S -> Subset of T means :: KURATO_2:def 11 for p being Point of T holds p in it iff for G being a_neighborhood of p ex k being Nat st for m being Nat st m > k holds S.m meets G; end; theorem :: KURATO_2:50 for S being SetSequence of the carrier of TOP-REAL n, p being Point of TOP-REAL n, p' being Point of Euclid n st p = p' holds p in Lim_inf S iff for r being real number st r > 0 ex k being Nat st for m being Nat st m > k holds S.m meets Ball (p', r); theorem :: KURATO_2:51 for T being non empty TopSpace, S being SetSequence of the carrier of T holds Cl Lim_inf S = Lim_inf S; theorem :: KURATO_2:52 for T being non empty TopSpace, S being SetSequence of the carrier of T holds Lim_inf S is closed; theorem :: KURATO_2:53 for T being non empty TopSpace, R, S being SetSequence of the carrier of T st R is subsequence of S holds Lim_inf S c= Lim_inf R; theorem :: KURATO_2:54 for T being non empty TopSpace, A, B being SetSequence of the carrier of T st for i being Nat holds A.i c= B.i holds Lim_inf A c= Lim_inf B; theorem :: KURATO_2:55 for T being non empty TopSpace, A, B, C being SetSequence of the carrier of T st for i being Nat holds C.i = A.i \/ B.i holds Lim_inf A \/ Lim_inf B c= Lim_inf C; theorem :: KURATO_2:56 for T being non empty TopSpace, A, B, C being SetSequence of the carrier of T st for i being Nat holds C.i = A.i /\ B.i holds Lim_inf C c= Lim_inf A /\ Lim_inf B; theorem :: KURATO_2:57 for T being non empty TopSpace, F, G being SetSequence of the carrier of T st for i being Nat holds G.i = Cl (F.i) holds Lim_inf G = Lim_inf F; theorem :: KURATO_2:58 for S being SetSequence of the carrier of TOP-REAL n, p being Point of TOP-REAL n holds (ex s being Real_Sequence of n st s is convergent & (for x being Nat holds s.x in S.x) & p = lim s) implies p in Lim_inf S; theorem :: KURATO_2:59 for T being non empty TopSpace, P being Subset of T, s being SetSequence of the carrier of T st (for i being Nat holds s.i c= P) holds Lim_inf s c= Cl P; theorem :: KURATO_2:60 for T being non empty TopSpace, F being SetSequence of the carrier of T, A being Subset of T st for i being Nat holds F.i = A holds Lim_inf F = Cl A; theorem :: KURATO_2:61 for T being non empty TopSpace, F being SetSequence of the carrier of T, A being closed Subset of T st for i being Nat holds F.i = A holds Lim_inf F = A; theorem :: KURATO_2:62 for S being SetSequence of the carrier of TOP-REAL n, P being Subset of TOP-REAL n st P is Bounded & (for i being Nat holds S.i c= P) holds Lim_inf S is Bounded; theorem :: KURATO_2:63 for S being SetSequence of the carrier of TOP-REAL 2, P being Subset of TOP-REAL 2 st P is Bounded & (for i being Nat holds S.i c= P) & for i being Nat holds S.i is compact holds Lim_inf S is compact; theorem :: KURATO_2:64 for A, B being SetSequence of the carrier of TOP-REAL n, C being SetSequence of the carrier of [: TOP-REAL n, TOP-REAL n :] st for i being Nat holds C.i = [:A.i, B.i:] holds [: Lim_inf A, Lim_inf B :] = Lim_inf C; theorem :: KURATO_2:65 for S being SetSequence of TOP-REAL 2 holds lim_inf S c= Lim_inf S; theorem :: KURATO_2:66 for C being Simple_closed_curve, i being Nat holds Fr (UBD L~Cage (C,i))` = L~Cage (C,i); begin :: Upper Topological Limit definition let T be non empty TopSpace; let S be SetSequence of the carrier of T; func Lim_sup S -> Subset of T means :: KURATO_2:def 12 for x being set holds x in it iff ex A being subsequence of S st x in Lim_inf A; end; theorem :: KURATO_2:67 for N being Nat, F being sequence of TOP-REAL N, x being Point of TOP-REAL N, x' being Point of Euclid N st x = x' holds x is_a_cluster_point_of F iff for r being real number, n being Nat st r > 0 holds ex m being Nat st n <= m & F.m in Ball (x', r); theorem :: KURATO_2:68 for T being non empty TopSpace, A being SetSequence of the carrier of T holds Lim_inf A c= Lim_sup A; theorem :: KURATO_2:69 for A, B, C being SetSequence of the carrier of TOP-REAL 2 st (for i being Nat holds A.i c= B.i) & C is subsequence of A holds ex D being subsequence of B st for i being Nat holds C.i c= D.i; theorem :: KURATO_2:70 for A, B, C being SetSequence of the carrier of TOP-REAL 2 st (for i being Nat holds A.i c= B.i) & C is subsequence of B holds ex D being subsequence of A st for i being Nat holds D.i c= C.i; theorem :: KURATO_2:71 :: (2) for A, B being SetSequence of the carrier of TOP-REAL 2 st for i being Nat holds A.i c= B.i holds Lim_sup A c= Lim_sup B; theorem :: KURATO_2:72 :: (3) for A, B, C being SetSequence of the carrier of TOP-REAL 2 st for i being Nat holds C.i = A.i \/ B.i holds Lim_sup A \/ Lim_sup B c= Lim_sup C; theorem :: KURATO_2:73 :: (4) for A, B, C being SetSequence of the carrier of TOP-REAL 2 st for i being Nat holds C.i = A.i /\ B.i holds Lim_sup C c= Lim_sup A /\ Lim_sup B; theorem :: KURATO_2:74 for A, B being SetSequence of the carrier of TOP-REAL 2, C, C1 being SetSequence of the carrier of [: TOP-REAL 2, TOP-REAL 2 :] st (for i being Nat holds C.i = [: A.i, B.i :]) & C1 is subsequence of C holds ex A1, B1 being SetSequence of the carrier of TOP-REAL 2 st A1 is subsequence of A & B1 is subsequence of B & for i being Nat holds C1.i = [: A1.i, B1.i :]; theorem :: KURATO_2:75 for A, B being SetSequence of the carrier of TOP-REAL 2, C being SetSequence of the carrier of [: TOP-REAL 2, TOP-REAL 2 :] st for i being Nat holds C.i = [: A.i, B.i :] holds Lim_sup C c= [: Lim_sup A, Lim_sup B :]; theorem :: KURATO_2:76 for T being non empty TopSpace, F being SetSequence of the carrier of T, A being Subset of T st for i being Nat holds F.i = A holds Lim_inf F = Lim_sup F; theorem :: KURATO_2:77 for F being SetSequence of the carrier of TOP-REAL 2, A being Subset of TOP-REAL 2 st for i being Nat holds F.i = A holds Lim_sup F = Cl A; theorem :: KURATO_2:78 for F, G being SetSequence of the carrier of TOP-REAL 2 st for i being Nat holds G.i = Cl (F.i) holds Lim_sup G = Lim_sup F;

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