Volume 9, 1997

University of Bialystok

Copyright (c) 1997 Association of Mizar Users

### The abstract of the Mizar article:

### Introduction to the Homotopy Theory

**by****Adam Grabowski**- Received September 10, 1997
- MML identifier: BORSUK_2

- [ Mizar article, MML identifier index ]

environ vocabulary PRE_TOPC, CARD_1, FUNCT_1, RELAT_1, SUBSET_1, BOOLE, COMPTS_1, ORDINAL2, FUNCT_4, FUNCOP_1, TOPS_2, BORSUK_1, GRAPH_1, RCOMP_1, RELAT_2, ARYTM_3, ARYTM_1, TREAL_1, SEQ_1, TOPMETR, SEQM_3, SETFAM_1, TARSKI, TSP_1, METRIC_1, URYSOHN1, PCOMPS_1, EUCLID, MCART_1, CONNSP_2, TOPS_1, BORSUK_2; notation TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, XREAL_0, NAT_1, REAL_1, RELAT_1, FUNCT_1, PARTFUN1, FUNCT_2, STRUCT_0, PRE_TOPC, EUCLID, TOPS_2, MCART_1, RCOMP_1, PCOMPS_1, CARD_1, TOPS_1, COMPTS_1, CONNSP_1, TREAL_1, FUNCT_4, SEQM_3, BORSUK_1, T_0TOPSP, TOPMETR, BINOP_1, FUNCT_3, GRCAT_1, URYSOHN1, TSP_1, WEIERSTR; constructors EUCLID, TOPS_2, CONNSP_1, REAL_1, T_0TOPSP, TREAL_1, SEQM_3, TOPS_1, WEIERSTR, RCOMP_1, COMPTS_1, YELLOW_8, URYSOHN1, GRCAT_1, MEMBERED; clusters SUBSET_1, STRUCT_0, TOPMETR, FUNCT_1, PRE_TOPC, BORSUK_1, TSP_1, YELLOW_6, YELLOW_8, WAYBEL10, METRIC_1, RELSET_1, MEMBERED, FUNCT_2; requirements REAL, NUMERALS, SUBSET, BOOLE, ARITHM; begin :: Preliminaries reserve T,T1,T2,S for non empty TopSpace; scheme FrCard { A() -> non empty set, X() -> set, F(set) -> set, P[set] }: Card { F(w) where w is Element of A(): w in X() & P[w] } <=` Card X(); theorem :: BORSUK_2:1 for f being map of T1,S, g being map of T2,S st T1 is SubSpace of T & T2 is SubSpace of T & ([#] T1) \/ ([#] T2) = [#] T & T1 is compact & T2 is compact & T is_T2 & f is continuous & g is continuous & ( for p be set st p in ([#] T1) /\ ([#] T2) holds f.p = g.p ) ex h being map of T,S st h = f+*g & h is continuous; definition let S, T be non empty TopSpace; cluster continuous map of S, T; end; definition let T be non empty TopStruct; cluster id T -> open continuous; end; definition let T be non empty TopStruct; cluster continuous one-to-one map of T, T; end; canceled; theorem :: BORSUK_2:3 for S, T being non empty TopSpace, f being map of S, T st f is_homeomorphism holds f" is open; begin :: Paths and arcwise connected spaces definition let T be TopStruct; let a, b be Point of T; given f be map of I[01], T such that f is continuous & f.0 = a & f.1 = b; mode Path of a, b -> map of I[01], T means :: BORSUK_2:def 1 it is continuous & it.0 = a & it.1 = b; end; theorem :: BORSUK_2:4 for T be non empty TopSpace, a be Point of T ex f be map of I[01], T st f is continuous & f.0 = a & f.1 = a; definition let T be non empty TopSpace; let a be Point of T; cluster continuous Path of a, a; end; definition let T be TopStruct; attr T is arcwise_connected means :: BORSUK_2:def 2 for a, b being Point of T ex f being map of I[01], T st f is continuous & f.0 = a & f.1 = b; end; definition cluster arcwise_connected non empty TopSpace; end; definition let T be arcwise_connected TopStruct; let a, b be Point of T; redefine mode Path of a, b means :: BORSUK_2:def 3 it is continuous & it.0 = a & it.1 = b; end; definition let T be arcwise_connected TopStruct; let a, b be Point of T; cluster -> continuous Path of a, b; end; reserve GY for non empty TopSpace, r,s for Real; theorem :: BORSUK_2:5 for GX being non empty TopSpace st GX is arcwise_connected holds GX is connected; definition cluster arcwise_connected -> connected (non empty TopSpace); end; begin :: Basic operations on paths definition let T be non empty TopSpace; let a, b, c be Point of T; let P be Path of a, b, Q be Path of b, c; given f, g be map of I[01], T such that f is continuous & f.0 = a & f.1 = b & g is continuous & g.0 = b & g.1 = c; func P + Q -> Path of a, c means :: BORSUK_2:def 4 for t being Point of I[01], t' being Real st t = t' holds ( 0 <= t' & t' <= 1/2 implies it.t = P.(2*t') ) & ( 1/2 <= t' & t' <= 1 implies it.t = Q.(2*t'-1) ); end; definition let T be non empty TopSpace; let a be Point of T; cluster constant Path of a, a; end; theorem :: BORSUK_2:6 for T being non empty TopSpace, a being Point of T, P being constant Path of a, a holds P = I[01] --> a; theorem :: BORSUK_2:7 for T being non empty TopSpace, a being Point of T, P being constant Path of a, a holds P + P = P; definition let T be non empty TopSpace, a be Point of T, P be constant Path of a, a; cluster P + P -> constant; end; definition let T be non empty TopSpace; let a, b be Point of T; let P be Path of a, b; given f be map of I[01], T such that f is continuous & f.0 = a & f.1 = b; func - P -> Path of b, a means :: BORSUK_2:def 5 for t being Point of I[01], t' being Real st t = t' holds it.t = P.(1-t'); end; theorem :: BORSUK_2:8 for T being non empty TopSpace, a being Point of T, P being constant Path of a, a holds - P = P; definition let T be non empty TopSpace, a be Point of T, P be constant Path of a, a; cluster - P -> constant; end; begin :: The product of two topological spaces theorem :: BORSUK_2:9 for X, Y being non empty TopSpace for A being Subset-Family of Y for f being map of X, Y holds f"(union A) = union (f"A); definition let S1, S2, T1, T2 be non empty TopSpace; let f be map of S1, S2, g be map of T1, T2; redefine func [:f, g:] -> map of [:S1, T1:], [:S2, T2:]; end; theorem :: BORSUK_2:10 for S1, S2, T1, T2 be non empty TopSpace, f be continuous map of S1, T1, g be continuous map of S2, T2, P1, P2 being Subset of [:T1, T2:] holds (P2 in Base-Appr P1 implies [:f,g:]"P2 is open); theorem :: BORSUK_2:11 for S1, S2, T1, T2 be non empty TopSpace, f be continuous map of S1, T1, g be continuous map of S2, T2, P2 being Subset of [:T1, T2:] holds (P2 is open implies [:f,g:]"P2 is open); theorem :: BORSUK_2:12 for S1, S2, T1, T2 be non empty TopSpace, f be continuous map of S1, T1, g be continuous map of S2, T2 holds [:f,g:] is continuous; definition cluster empty -> T_0 TopStruct; end; definition let T1, T2 be discerning (non empty TopSpace); cluster [:T1, T2:] -> discerning; end; canceled; theorem :: BORSUK_2:14 for T1, T2 be non empty TopSpace holds T1 is_T1 & T2 is_T1 implies [:T1, T2:] is_T1; definition let T1, T2 be being_T1 (non empty TopSpace); cluster [:T1, T2:] -> being_T1; end; definition let T1, T2 be being_T2 (non empty TopSpace); cluster [:T1, T2:] -> being_T2; end; definition cluster I[01] -> compact being_T2; end; definition let n be Nat; cluster TOP-REAL n -> being_T2; end; definition let T be non empty arcwise_connected TopSpace; let a, b be Point of T; let P, Q be Path of a, b; pred P, Q are_homotopic means :: BORSUK_2:def 6 ex f being map of [:I[01],I[01]:], T st f is continuous & for s being Point of I[01] holds f.(s,0) = P.s & f.(s,1) = Q.s & for t being Point of I[01] holds f.(0,t) = a & f.(1,t) = b; reflexivity; symmetry; end;

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