Volume 11, 1999

University of Bialystok

Copyright (c) 1999 Association of Mizar Users

### The abstract of the Mizar article:

### Properties of the Product of Compact Topological Spaces

**by****Adam Grabowski**- Received February 13, 1999
- MML identifier: BORSUK_3

- [ Mizar article, MML identifier index ]

environ vocabulary PRE_TOPC, SUBSET_1, FUNCOP_1, ORDINAL2, RELAT_1, FUNCT_1, TOPS_2, T_0TOPSP, BOOLE, COMPTS_1, FUNCT_3, PARTFUN1, PBOOLE, TARSKI, BORSUK_1, TOPS_1, CONNSP_2, SETFAM_1, FINSET_1; notation TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, RELAT_1, SETFAM_1, FUNCT_1, STRUCT_0, PRE_TOPC, TOPS_2, TOPS_1, COMPTS_1, BORSUK_1, T_0TOPSP, PBOOLE, FUNCT_2, FINSET_1, FUNCT_3, CAT_1, CONNSP_2, GRCAT_1; constructors TOPS_2, T_0TOPSP, TOPS_1, COMPTS_1, WAYBEL_3, GRCAT_1, BORSUK_1, MEMBERED; clusters STRUCT_0, PRE_TOPC, BORSUK_1, BORSUK_2, TOPS_1, WAYBEL_3, WAYBEL12, PSCOMP_1, RELSET_1, SUBSET_1, XBOOLE_0; requirements SUBSET, BOOLE; begin :: Preliminaries theorem :: BORSUK_3:1 for S, T being TopSpace holds [#][:S, T:] = [:[#]S, [#]T:]; definition let X be set, Y be empty set; cluster [:X, Y:] -> empty; end; definition let X be empty set, Y be set; cluster [:X, Y:] -> empty; end; theorem :: BORSUK_3:2 for X, Y being non empty TopSpace, x being Point of X holds Y --> x is continuous map of Y, X|{x}; definition let T be non empty TopStruct; cluster id T -> being_homeomorphism; end; definition let S, T be non empty TopStruct; redefine pred S, T are_homeomorphic; reflexivity; symmetry; end; theorem :: BORSUK_3:3 for S, T, V being non empty TopSpace st S, T are_homeomorphic & T, V are_homeomorphic holds S, V are_homeomorphic; begin :: On the projections and empty topological spaces definition let T be TopStruct, P be empty Subset of T; cluster T | P -> empty; end; definition cluster strict empty TopSpace; end; theorem :: BORSUK_3:4 for T1 being TopSpace, T2 being empty TopSpace holds [:T1, T2:] is empty & [:T2, T1:] is empty; theorem :: BORSUK_3:5 for T being empty TopSpace holds T is compact; definition cluster empty -> compact TopSpace; end; definition let T1 be TopSpace, T2 be empty TopSpace; cluster [:T1, T2:] -> empty; end; theorem :: BORSUK_3:6 for X, Y being non empty TopSpace, x being Point of X, f being map of [:Y, X | {x}:], Y st f = pr1(the carrier of Y, {x}) holds f is one-to-one; theorem :: BORSUK_3:7 for X, Y being non empty TopSpace, x being Point of X, f being map of [:X | {x}, Y:], Y st f = pr2({x}, the carrier of Y) holds f is one-to-one; theorem :: BORSUK_3:8 for X, Y being non empty TopSpace, x being Point of X, f being map of [:Y, X | {x}:], Y st f = pr1(the carrier of Y, {x}) holds f" = <:id Y, Y --> x:>; theorem :: BORSUK_3:9 for X, Y being non empty TopSpace, x being Point of X, f being map of [:X | {x}, Y:], Y st f = pr2({x}, the carrier of Y) holds f" = <:Y --> x, id Y:>; theorem :: BORSUK_3:10 for X, Y being non empty TopSpace, x being Point of X, f being map of [:Y, X | {x}:], Y st f = pr1(the carrier of Y, {x}) holds f is_homeomorphism; theorem :: BORSUK_3:11 for X, Y being non empty TopSpace, x being Point of X, f being map of [:X | {x}, Y:], Y st f = pr2({x}, the carrier of Y) holds f is_homeomorphism; begin :: On the product of compact spaces theorem :: BORSUK_3:12 for X being non empty TopSpace, Y being compact non empty TopSpace, G being open Subset of [:X, Y:], x being set st x in { x' where x' is Point of X : [:{x'}, the carrier of Y:] c= G } ex f being ManySortedSet of the carrier of Y st for i being set st i in the carrier of Y ex G1 being Subset of X, H1 being Subset of Y st f.i = [G1,H1] & [x, i] in [:G1, H1:] & G1 is open & H1 is open & [:G1, H1:] c= G; theorem :: BORSUK_3:13 for X being non empty TopSpace, Y being compact non empty TopSpace, G being open Subset of [:Y, X:] holds for x being set st x in { y where y is Point of X : [:[#]Y, {y}:] c= G } holds ex R be open Subset of X st x in R & R c= { y where y is Point of X : [:[#]Y, {y}:] c= G }; theorem :: BORSUK_3:14 for X being non empty TopSpace, Y being compact non empty TopSpace, G being open Subset of [:Y, X:] holds { x where x is Point of X : [:[#]Y, {x}:] c= G } in the topology of X; theorem :: BORSUK_3:15 for X, Y being non empty TopSpace, x being Point of X holds [: X | {x}, Y :], Y are_homeomorphic; theorem :: BORSUK_3:16 for S, T being non empty TopSpace st S, T are_homeomorphic & S is compact holds T is compact; theorem :: BORSUK_3:17 for X, Y being TopSpace, XV being SubSpace of X holds [:Y, XV:] is SubSpace of [:Y, X:]; theorem :: BORSUK_3:18 for X being non empty TopSpace, Y being compact non empty TopSpace, x being Point of X, Z being Subset of [:Y, X:] st Z = [:[#]Y, {x}:] holds Z is compact; theorem :: BORSUK_3:19 for X being non empty TopSpace, Y being compact non empty TopSpace, x being Point of X holds [:Y, X|{x}:] is compact; theorem :: BORSUK_3:20 for X, Y being compact non empty TopSpace, R being Subset-Family of X st R = { Q where Q is open Subset of X : [:[#]Y, Q:] c= union Base-Appr [#][:Y, X:] } holds R is open & R is_a_cover_of [#]X; theorem :: BORSUK_3:21 for X, Y being compact non empty TopSpace, R being Subset-Family of X, F being Subset-Family of [:Y, X:] st F is_a_cover_of [:Y, X:] & F is open & R = { Q where Q is open Subset of X : ex FQ being Subset-Family of [:Y, X:] st FQ c= F & FQ is finite & [:[#]Y, Q:] c= union FQ } holds R is open & R is_a_cover_of X; theorem :: BORSUK_3:22 for X, Y being compact non empty TopSpace, R being Subset-Family of X, F being Subset-Family of [:Y, X:] st F is_a_cover_of [:Y, X:] & F is open & R = { Q where Q is open Subset of X : ex FQ being Subset-Family of [:Y, X:] st FQ c= F & FQ is finite & [:[#]Y, Q:] c= union FQ } holds ex C being Subset-Family of X st C c= R & C is finite & C is_a_cover_of X; theorem :: BORSUK_3:23 for X, Y being compact non empty TopSpace, F being Subset-Family of [:Y, X:] st F is_a_cover_of [:Y, X:] & F is open ex G being Subset-Family of [:Y, X:] st G c= F & G is_a_cover_of [:Y, X:] & G is finite; theorem :: BORSUK_3:24 for T1, T2 be TopSpace st T1 is compact & T2 is compact holds [:T1, T2:] is compact; definition let T1, T2 be compact TopSpace; cluster [:T1, T2:] -> compact; end; theorem :: BORSUK_3:25 for X, Y being non empty TopSpace, XV being non empty SubSpace of X, YV being non empty SubSpace of Y holds [:XV, YV:] is SubSpace of [:X, Y:]; theorem :: BORSUK_3:26 for X, Y being non empty TopSpace, Z being non empty Subset of [:Y, X:], V being non empty Subset of X, W being non empty Subset of Y st Z = [:W, V:] holds the TopStruct of [:Y | W, X | V:] = the TopStruct of [:Y, X:] | Z; definition let T be TopSpace; cluster compact Subset of T; end; definition let T be TopSpace, P be compact Subset of T; cluster T | P -> compact; end; theorem :: BORSUK_3:27 for T1, T2 being TopSpace, S1 being Subset of T1, S2 being Subset of T2 st S1 is compact & S2 is compact holds [:S1, S2:] is compact Subset of [:T1, T2:];

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