:: Properties of the Product of Compact Topological Spaces
::
:: Copyright (c) 1999-2019 Association of Mizar Users

theorem :: BORSUK_3:1
for S, T being TopSpace holds [#] [:S,T:] = [:([#] S),([#] T):] by BORSUK_1:def 2;

theorem Th2: :: BORSUK_3:2
for X, Y being non empty TopSpace
for x being Point of X holds Y --> x is continuous Function of Y,(X | {x})
proof end;

registration
let T be TopStruct ;
coherence by TOPGRP_1:20;
end;

Lm1: for S being TopStruct holds S,S are_homeomorphic
proof end;

Lm2: for S, T being non empty TopStruct st S,T are_homeomorphic holds
T,S are_homeomorphic

proof end;

definition
let S, T be TopStruct ;
:: original: are_homeomorphic
redefine pred S,T are_homeomorphic ;
reflexivity
for S being TopStruct holds R37(b1,b1)
by Lm1;
end;

definition
let S, T be non empty TopStruct ;
:: original: are_homeomorphic
redefine pred S,T are_homeomorphic ;
reflexivity
for S being non empty TopStruct holds R37(b1,b1)
by Lm1;
symmetry
for S, T being non empty TopStruct st R37(b1,b2) holds
R37(b2,b1)
by Lm2;
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
proof end;

registration
let T be TopStruct ;
let P be empty Subset of T;
cluster T | P -> empty ;
coherence
T | P is empty
;
end;

registration
coherence
for b1 being TopSpace st b1 is empty holds
b1 is compact
;
end;

theorem Th4: :: BORSUK_3:4
for X, Y being non empty TopSpace
for x being Point of X
for f being Function of [:Y,(X | {x}):],Y st f = pr1 ( the carrier of Y,{x}) holds
f is one-to-one
proof end;

theorem Th5: :: BORSUK_3:5
for X, Y being non empty TopSpace
for x being Point of X
for f being Function of [:(X | {x}),Y:],Y st f = pr2 ({x}, the carrier of Y) holds
f is one-to-one
proof end;

theorem Th6: :: BORSUK_3:6
for X, Y being non empty TopSpace
for x being Point of X
for f being Function of [:Y,(X | {x}):],Y st f = pr1 ( the carrier of Y,{x}) holds
f " = <:(id Y),(Y --> x):>
proof end;

theorem Th7: :: BORSUK_3:7
for X, Y being non empty TopSpace
for x being Point of X
for f being Function of [:(X | {x}),Y:],Y st f = pr2 ({x}, the carrier of Y) holds
f " = <:(Y --> x),(id Y):>
proof end;

theorem :: BORSUK_3:8
for X, Y being non empty TopSpace
for x being Point of X
for f being Function of [:Y,(X | {x}):],Y st f = pr1 ( the carrier of Y,{x}) holds
f is being_homeomorphism
proof end;

theorem Th9: :: BORSUK_3:9
for X, Y being non empty TopSpace
for x being Point of X
for f being Function of [:(X | {x}),Y:],Y st f = pr2 ({x}, the carrier of Y) holds
f is being_homeomorphism
proof end;

theorem :: BORSUK_3:10
for X being non empty TopSpace
for Y being non empty compact TopSpace
for G being open Subset of [:X,Y:]
for x being set st [:{x}, the carrier of Y:] c= G holds
ex f being ManySortedSet of the carrier of Y st
for i being object st i in the carrier of Y holds
ex G1 being Subset of X ex 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 )
proof end;

theorem Th11: :: BORSUK_3:11
for X being non empty TopSpace
for Y being non empty compact TopSpace
for G being open Subset of [:Y,X:]
for x being set st [:([#] Y),{x}:] c= G holds
ex R being open Subset of X st
( x in R & R c= { y where y is Point of X : [:([#] Y),{y}:] c= G } )
proof end;

theorem Th12: :: BORSUK_3:12
for X being non empty TopSpace
for Y being non empty compact TopSpace
for G being open Subset of [:Y,X:] holds { x where x is Point of X : [:([#] Y),{x}:] c= G } in the topology of X
proof end;

theorem Th13: :: BORSUK_3:13
for X, Y being non empty TopSpace
for x being Point of X holds [:(X | {x}),Y:],Y are_homeomorphic
proof end;

Lm3: for X, Y being non empty TopSpace
for x being Point of X
for Z being non empty Subset of X st Z = {x} holds
[:Y,(X | Z):],Y are_homeomorphic

proof end;

theorem Th14: :: BORSUK_3:14
for S, T being non empty TopSpace st S,T are_homeomorphic & S is compact holds
T is compact
proof end;

theorem Th15: :: BORSUK_3:15
for X, Y being TopSpace
for XV being SubSpace of X holds [:Y,XV:] is SubSpace of [:Y,X:]
proof end;

Lm4: for X, Y being TopSpace
for Z being Subset of [:Y,X:]
for V being Subset of X st Z = [:([#] Y),V:] holds
TopStruct(# the carrier of [:Y,(X | V):], the topology of [:Y,(X | V):] #) = TopStruct(# the carrier of ([:Y,X:] | Z), the topology of ([:Y,X:] | Z) #)

proof end;

theorem Th16: :: BORSUK_3:16
for X being non empty TopSpace
for Y being non empty compact TopSpace
for x being Point of X
for Z being Subset of [:Y,X:] st Z = [:([#] Y),{x}:] holds
Z is compact
proof end;

registration
let X be non empty TopSpace;
let Y be non empty compact TopSpace;
let x be Point of X;
cluster [:Y,(X | {x}):] -> compact ;
coherence
[:Y,(X | {x}):] is compact
proof end;
end;

theorem :: BORSUK_3:17
for X, Y being non empty compact TopSpace
for 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 Cover of [#] X )
proof end;

theorem Th18: :: BORSUK_3:18
for X, Y being non empty compact TopSpace
for R being Subset-Family of X
for F being Subset-Family of [:Y,X:] st F is 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 Cover of X )
proof end;

theorem Th19: :: BORSUK_3:19
for X, Y being non empty compact TopSpace
for R being Subset-Family of X
for F being Subset-Family of [:Y,X:] st F is 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 Cover of X )
proof end;

theorem Th20: :: BORSUK_3:20
for X, Y being non empty compact TopSpace
for F being Subset-Family of [:Y,X:] st F is Cover of [:Y,X:] & F is open holds
ex G being Subset-Family of [:Y,X:] st
( G c= F & G is Cover of [:Y,X:] & G is finite )
proof end;

Lm5: for T1, T2 being non empty compact TopSpace holds [:T1,T2:] is compact
by Th20;

registration
let T1, T2 be compact TopSpace;
cluster [:T1,T2:] -> compact ;
coherence
[:T1,T2:] is compact
proof end;
end;

Lm6: for X, Y being TopSpace
for XV being SubSpace of X holds [:XV,Y:] is SubSpace of [:X,Y:]

proof end;

theorem Th21: :: BORSUK_3:21
for X, Y being TopSpace
for XV being SubSpace of X
for YV being SubSpace of Y holds [:XV,YV:] is SubSpace of [:X,Y:]
proof end;

theorem Th22: :: BORSUK_3:22
for X, Y being TopSpace
for Z being Subset of [:Y,X:]
for V being Subset of X
for W being Subset of Y st Z = [:W,V:] holds
TopStruct(# the carrier of [:(Y | W),(X | V):], the topology of [:(Y | W),(X | V):] #) = TopStruct(# the carrier of ([:Y,X:] | Z), the topology of ([:Y,X:] | Z) #)
proof end;

registration
let T be TopSpace;
cluster empty for Element of bool the carrier of T;
existence
ex b1 being Subset of T st b1 is empty
proof end;
end;

registration
let T be TopSpace;
let P be compact Subset of T;
cluster T | P -> compact ;
coherence
T | P is compact
proof end;
end;

theorem :: BORSUK_3:23
for T1, T2 being TopSpace
for S1 being Subset of T1
for S2 being Subset of T2 st S1 is compact & S2 is compact holds
[:S1,S2:] is compact Subset of [:T1,T2:]
proof end;