:: A Borsuk Theorem on Homotopy Types
:: by Andrzej Trybulec
::
:: Received August 1, 1991
:: Copyright (c) 1991-2021 Association of Mizar Users

:: Topological preliminaries

theorem Th1: :: BORSUK_1:1

for X being TopStruct
for Y being SubSpace of X holds the carrier of Y c= the carrier of X

proof end;

definition

let X, Y be non empty TopSpace;
let F be Function of X,Y;

redefine attr F is continuous means :: BORSUK_1:def 1
for W being Point of X
for G being a_neighborhood of F . W ex H being a_neighborhood of W st F .: H c= G;

compatibility
( F is continuous iff for W being Point of X
for G being a_neighborhood of F . W ex H being a_neighborhood of W st F .: H c= G )

proof end;

end;

:: deftheorem defines continuous BORSUK_1:def 1 :
for X, Y being non empty TopSpace
for F being Function of X,Y holds
( F is continuous iff for W being Point of X
for G being a_neighborhood of F . W ex H being a_neighborhood of W st F .: H c= G );

registration

let X, Y, Z be non empty TopSpace;
let F be continuous Function of X,Y;
let G be continuous Function of Y,Z;

cluster F * G -> continuous for Function of X,Z;

coherence
for b₁ being Function of X,Z st b₁ = G * F holds
b₁ is continuous by TOPS_2:46;

end;

theorem Th2: :: BORSUK_1:2

for X, Y being non empty TopSpace
for A being continuous Function of X,Y
for G being Subset of Y holds A " (Int G) c= Int (A " G)

proof end;

theorem Th3: :: BORSUK_1:3

for X, Y being non empty TopSpace
for W being Point of Y
for A being continuous Function of X,Y
for G being a_neighborhood of W holds A " G is a_neighborhood of A " {W}

proof end;

definition

let X, Y be non empty TopSpace;
let W be Point of Y;
let A be continuous Function of X,Y;
let G be a_neighborhood of W;
:: original: "
redefine func A " G -> a_neighborhood of A " {W};
correctness
coherence
A " G is a_neighborhood of A " {W};
by Th3;

end;

theorem Th4: :: BORSUK_1:4

for X being non empty TopSpace
for A, B being Subset of X
for U being a_neighborhood of B st A c= B holds
U is a_neighborhood of A

proof end;

registration

let X be non empty TopSpace;
let x be Point of X;

cluster {x} -> compact for Subset of X;

coherence
for b₁ being Subset of X st b₁ = {x} holds
b₁ is compact

proof end;

end;

::
:: Cartesian products of topological spaces
::

definition

let X, Y be TopSpace;

func [:X,Y:] -> strict TopSpace means :Def2: :: BORSUK_1:def 2
( the carrier of it = [: the carrier of X, the carrier of Y:] & the topology of it = { (union A) where A is Subset-Family of it : A c= { [:X1,Y1:] where X1 is Subset of X, Y1 is Subset of Y : ( X1 in the topology of X & Y1 in the topology of Y ) } } );

existence
ex b₁ being strict TopSpace st
( the carrier of b₁ = [: the carrier of X, the carrier of Y:] & the topology of b₁ = { (union A) where A is Subset-Family of b₁ : A c= { [:X1,Y1:] where X1 is Subset of X, Y1 is Subset of Y : ( X1 in the topology of X & Y1 in the topology of Y ) } } )

proof end;

uniqueness
for b₁, b₂ being strict TopSpace st the carrier of b₁ = [: the carrier of X, the carrier of Y:] & the topology of b₁ = { (union A) where A is Subset-Family of b₁ : A c= { [:X1,Y1:] where X1 is Subset of X, Y1 is Subset of Y : ( X1 in the topology of X & Y1 in the topology of Y ) } } & the carrier of b₂ = [: the carrier of X, the carrier of Y:] & the topology of b₂ = { (union A) where A is Subset-Family of b₂ : A c= { [:X1,Y1:] where X1 is Subset of X, Y1 is Subset of Y : ( X1 in the topology of X & Y1 in the topology of Y ) } } holds
b₁ = b₂ ;

end;

:: deftheorem Def2 defines [: BORSUK_1:def 2 :
for X, Y being TopSpace
for b₃ being strict TopSpace holds
( b₃ = [:X,Y:] iff ( the carrier of b₃ = [: the carrier of X, the carrier of Y:] & the topology of b₃ = { (union A) where A is Subset-Family of b₃ : A c= { [:X1,Y1:] where X1 is Subset of X, Y1 is Subset of Y : ( X1 in the topology of X & Y1 in the topology of Y ) } } ) );

registration

let T1 be TopSpace;
let T2 be empty TopSpace;

cluster [:T1,T2:] -> empty strict ;

coherence
[:T1,T2:] is empty

proof end;

cluster [:T2,T1:] -> empty strict ;

coherence
[:T2,T1:] is empty

proof end;

end;

registration

let X, Y be non empty TopSpace;

cluster [:X,Y:] -> non empty strict ;

coherence
not [:X,Y:] is empty

proof end;

end;

theorem Th5: :: BORSUK_1:5

for X, Y being TopSpace
for B being Subset of [:X,Y:] holds
( B is open iff ex A being Subset-Family of [:X,Y:] st
( B = union A & ( for e being set st e in A holds
ex X1 being Subset of X ex Y1 being Subset of Y st
( e = [:X1,Y1:] & X1 is open & Y1 is open ) ) ) )

proof end;

definition

let X, Y be TopSpace;
let A be Subset of X;
let B be Subset of Y;
:: original: [:
redefine func [:A,B:] -> Subset of [:X,Y:];
correctness
coherence
[:A,B:] is Subset of [:X,Y:];

proof end;

end;

definition

let X, Y be non empty TopSpace;
let x be Point of X;
let y be Point of Y;
:: original: [
redefine func [x,y] -> Point of [:X,Y:];
correctness
coherence
[x,y] is Point of [:X,Y:];

proof end;

end;

theorem Th6: :: BORSUK_1:6

for X, Y being TopSpace
for V being Subset of X
for W being Subset of Y st V is open & W is open holds
[:V,W:] is open

proof end;

theorem Th7: :: BORSUK_1:7

for X, Y being TopSpace
for V being Subset of X
for W being Subset of Y holds Int [:V,W:] = [:(Int V),(Int W):]

proof end;

theorem Th8: :: BORSUK_1:8

for X, Y being non empty TopSpace
for x being Point of X
for y being Point of Y
for V being a_neighborhood of x
for W being a_neighborhood of y holds [:V,W:] is a_neighborhood of [x,y]

proof end;

theorem Th9: :: BORSUK_1:9

for X, Y being non empty TopSpace
for A being Subset of X
for B being Subset of Y
for V being a_neighborhood of A
for W being a_neighborhood of B holds [:V,W:] is a_neighborhood of [:A,B:]

proof end;

definition

let X, Y be non empty TopSpace;
let x be Point of X;
let y be Point of Y;
let V be a_neighborhood of x;
let W be a_neighborhood of y;
:: original: [:
redefine func [:V,W:] -> a_neighborhood of [x,y];
correctness
coherence
[:V,W:] is a_neighborhood of [x,y];
by Th8;

end;

theorem Th10: :: BORSUK_1:10

for X, Y being non empty TopSpace
for XT being Point of [:X,Y:] ex W being Point of X ex T being Point of Y st XT = [W,T]

proof end;

definition

let X, Y be non empty TopSpace;
let A be Subset of X;
let t be Point of Y;
let V be a_neighborhood of A;
let W be a_neighborhood of t;
:: original: [:
redefine func [:V,W:] -> a_neighborhood of [:A,{t}:];
correctness
coherence
[:V,W:] is a_neighborhood of [:A,{t}:];

proof end;

end;

definition

let X, Y be TopSpace;
let A be Subset of [:X,Y:];

func Base-Appr A -> Subset-Family of [:X,Y:] equals :: BORSUK_1:def 3
{ [:X1,Y1:] where X1 is Subset of X, Y1 is Subset of Y : ( [:X1,Y1:] c= A & X1 is open & Y1 is open ) } ;

coherence
{ [:X1,Y1:] where X1 is Subset of X, Y1 is Subset of Y : ( [:X1,Y1:] c= A & X1 is open & Y1 is open ) } is Subset-Family of [:X,Y:]

proof end;

end;

:: deftheorem defines Base-Appr BORSUK_1:def 3 :
for X, Y being TopSpace
for A being Subset of [:X,Y:] holds Base-Appr A = { [:X1,Y1:] where X1 is Subset of X, Y1 is Subset of Y : ( [:X1,Y1:] c= A & X1 is open & Y1 is open ) } ;

registration

let X, Y be TopSpace;
let A be Subset of [:X,Y:];

cluster Base-Appr A -> open ;

coherence
Base-Appr A is open

proof end;

end;

theorem Th11: :: BORSUK_1:11

for X, Y being TopSpace
for A, B being Subset of [:X,Y:] st A c= B holds
Base-Appr A c= Base-Appr B

proof end;

theorem Th12: :: BORSUK_1:12

for X, Y being TopSpace
for A being Subset of [:X,Y:] holds union (Base-Appr A) c= A

proof end;

theorem Th13: :: BORSUK_1:13

for X, Y being TopSpace
for A being Subset of [:X,Y:] st A is open holds
A = union (Base-Appr A)

proof end;

theorem Th14: :: BORSUK_1:14

for X, Y being TopSpace
for A being Subset of [:X,Y:] holds Int A = union (Base-Appr A)

proof end;

definition

let X, Y be non empty TopSpace;

func Pr1 (X,Y) -> Function of (bool the carrier of [:X,Y:]),(bool the carrier of X) equals :: BORSUK_1:def 4
.: (pr1 ( the carrier of X, the carrier of Y));

coherence
.: (pr1 ( the carrier of X, the carrier of Y)) is Function of (bool the carrier of [:X,Y:]),(bool the carrier of X)

proof end;

correctness
;

func Pr2 (X,Y) -> Function of (bool the carrier of [:X,Y:]),(bool the carrier of Y) equals :: BORSUK_1:def 5
.: (pr2 ( the carrier of X, the carrier of Y));

coherence
.: (pr2 ( the carrier of X, the carrier of Y)) is Function of (bool the carrier of [:X,Y:]),(bool the carrier of Y)

proof end;

correctness
;

end;

:: deftheorem defines Pr1 BORSUK_1:def 4 :
for X, Y being non empty TopSpace holds Pr1 (X,Y) = .: (pr1 ( the carrier of X, the carrier of Y));

:: deftheorem defines Pr2 BORSUK_1:def 5 :
for X, Y being non empty TopSpace holds Pr2 (X,Y) = .: (pr2 ( the carrier of X, the carrier of Y));

theorem Th15: :: BORSUK_1:15

for X, Y being non empty TopSpace
for A being Subset of [:X,Y:]
for H being Subset-Family of [:X,Y:] st ( for e being set st e in H holds
( e c= A & ex X1 being Subset of X ex Y1 being Subset of Y st e = [:X1,Y1:] ) ) holds
[:(union ((Pr1 (X,Y)) .: H)),(meet ((Pr2 (X,Y)) .: H)):] c= A

proof end;

theorem Th16: :: BORSUK_1:16

for X, Y being non empty TopSpace
for H being Subset-Family of [:X,Y:]
for C being set st C in (Pr1 (X,Y)) .: H holds
ex D being Subset of [:X,Y:] st
( D in H & C = (pr1 ( the carrier of X, the carrier of Y)) .: D )

proof end;

theorem Th17: :: BORSUK_1:17

for X, Y being non empty TopSpace
for H being Subset-Family of [:X,Y:]
for C being set st C in (Pr2 (X,Y)) .: H holds
ex D being Subset of [:X,Y:] st
( D in H & C = (pr2 ( the carrier of X, the carrier of Y)) .: D )

proof end;

theorem Th18: :: BORSUK_1:18

for X, Y being non empty TopSpace
for D being Subset of [:X,Y:] st D is open holds
for X1 being Subset of X
for Y1 being Subset of Y holds
( ( X1 = (pr1 ( the carrier of X, the carrier of Y)) .: D implies X1 is open ) & ( Y1 = (pr2 ( the carrier of X, the carrier of Y)) .: D implies Y1 is open ) )

proof end;

theorem Th19: :: BORSUK_1:19

for X, Y being non empty TopSpace
for H being Subset-Family of [:X,Y:] st H is open holds
( (Pr1 (X,Y)) .: H is open & (Pr2 (X,Y)) .: H is open )

proof end;

theorem Th20: :: BORSUK_1:20

for X, Y being non empty TopSpace
for H being Subset-Family of [:X,Y:] st ( (Pr1 (X,Y)) .: H = {} or (Pr2 (X,Y)) .: H = {} ) holds
H = {}

proof end;

theorem Th21: :: BORSUK_1:21

for X, Y being non empty TopSpace
for H being Subset-Family of [:X,Y:]
for X1 being Subset of X
for Y1 being Subset of Y st H is Cover of [:X1,Y1:] holds
( ( Y1 <> {} implies (Pr1 (X,Y)) .: H is Cover of X1 ) & ( X1 <> {} implies (Pr2 (X,Y)) .: H is Cover of Y1 ) )

proof end;

theorem Th22: :: BORSUK_1:22

for X, Y being TopSpace
for H being Subset-Family of X
for Y being Subset of X st H is Cover of Y holds
ex F being Subset-Family of X st
( F c= H & F is Cover of Y & ( for C being set st C in F holds
C meets Y ) )

proof end;

theorem Th23: :: BORSUK_1:23

for X, Y being non empty TopSpace
for F being Subset-Family of X
for H being Subset-Family of [:X,Y:] st F is finite & F c= (Pr1 (X,Y)) .: H holds
ex G being Subset-Family of [:X,Y:] st
( G c= H & G is finite & F = (Pr1 (X,Y)) .: G )

proof end;

theorem :: BORSUK_1:24

for X, Y being non empty TopSpace
for X1 being Subset of X
for Y1 being Subset of Y st [:X1,Y1:] <> {} holds
( (Pr1 (X,Y)) . [:X1,Y1:] = X1 & (Pr2 (X,Y)) . [:X1,Y1:] = Y1 ) by EQREL_1:50;

theorem Th25: :: BORSUK_1:25

for X, Y being non empty TopSpace
for t being Point of Y
for A being Subset of X st A is compact holds
for G being a_neighborhood of [:A,{t}:] ex V being a_neighborhood of A ex W being a_neighborhood of t st [:V,W:] c= G

proof end;

::
:: Partitions of topological spaces
::

definition

let X be 1-sorted ;

func TrivDecomp X -> a_partition of the carrier of X equals :: BORSUK_1:def 6
Class (id the carrier of X);

coherence
Class (id the carrier of X) is a_partition of the carrier of X ;

end;

:: deftheorem defines TrivDecomp BORSUK_1:def 6 :
for X being 1-sorted holds TrivDecomp X = Class (id the carrier of X);

registration

let X be non empty 1-sorted ;

cluster TrivDecomp X -> non empty ;

coherence
not TrivDecomp X is empty ;

end;

theorem Th26: :: BORSUK_1:26

for X being non empty TopSpace
for A being Subset of X st A in TrivDecomp X holds
ex x being Point of X st A = {x}

proof end;

Lm1: for X being non empty TopSpace
for A being Subset of X
for V being a_neighborhood of A ex W being Subset of X st
( W is open & A c= W & W c= V & ( for B being Subset of X st B in TrivDecomp X & B meets W holds
B c= W ) )

proof end;

definition

let X be TopSpace;
let D be a_partition of the carrier of X;

func space D -> strict TopSpace means :Def7: :: BORSUK_1:def 7
( the carrier of it = D & the topology of it = { A where A is Subset of D : union A in the topology of X } );

existence
ex b₁ being strict TopSpace st
( the carrier of b₁ = D & the topology of b₁ = { A where A is Subset of D : union A in the topology of X } )

proof end;

uniqueness
for b₁, b₂ being strict TopSpace st the carrier of b₁ = D & the topology of b₁ = { A where A is Subset of D : union A in the topology of X } & the carrier of b₂ = D & the topology of b₂ = { A where A is Subset of D : union A in the topology of X } holds
b₁ = b₂ ;

end;

:: deftheorem Def7 defines space BORSUK_1:def 7 :
for X being TopSpace
for D being a_partition of the carrier of X
for b₃ being strict TopSpace holds
( b₃ = space D iff ( the carrier of b₃ = D & the topology of b₃ = { A where A is Subset of D : union A in the topology of X } ) );

registration

let X be non empty TopSpace;
let D be a_partition of the carrier of X;

cluster space D -> non empty strict ;

coherence
not space D is empty by Def7;

end;

theorem Th27: :: BORSUK_1:27

for X being non empty TopSpace
for D being non empty a_partition of the carrier of X
for A being Subset of D holds
( union A in the topology of X iff A in the topology of (space D) )

proof end;

definition

let X be non empty TopSpace;
let D be non empty a_partition of the carrier of X;

func Proj D -> continuous Function of X,(space D) equals :: BORSUK_1:def 8
proj D;

coherence
proj D is continuous Function of X,(space D)

proof end;

correctness
;

end;

:: deftheorem defines Proj BORSUK_1:def 8 :
for X being non empty TopSpace
for D being non empty a_partition of the carrier of X holds Proj D = proj D;

theorem :: BORSUK_1:28

for X being non empty TopSpace
for D being non empty a_partition of the carrier of X
for W being Point of X holds W in (Proj D) . W by EQREL_1:def 9;

theorem Th29: :: BORSUK_1:29

for X being non empty TopSpace
for D being non empty a_partition of the carrier of X
for W being Point of (space D) ex W9 being Point of X st (Proj D) . W9 = W

proof end;

theorem Th30: :: BORSUK_1:30

for X being non empty TopSpace
for D being non empty a_partition of the carrier of X holds rng (Proj D) = the carrier of (space D)

proof end;

definition

let XX be non empty TopSpace;
let X be non empty SubSpace of XX;
let D be non empty a_partition of the carrier of X;

func TrivExt D -> non empty a_partition of the carrier of XX equals :: BORSUK_1:def 9
D \/ { {p} where p is Point of XX : not p in the carrier of X } ;

coherence
D \/ { {p} where p is Point of XX : not p in the carrier of X } is non empty a_partition of the carrier of XX

proof end;

correctness
;

end;

:: deftheorem defines TrivExt BORSUK_1:def 9 :
for XX being non empty TopSpace
for X being non empty SubSpace of XX
for D being non empty a_partition of the carrier of X holds TrivExt D = D \/ { {p} where p is Point of XX : not p in the carrier of X } ;

theorem Th31: :: BORSUK_1:31

for XX being non empty TopSpace
for X being non empty SubSpace of XX
for D being non empty a_partition of the carrier of X
for A being Subset of XX holds
( not A in TrivExt D or A in D or ex x being Point of XX st
( not x in [#] X & A = {x} ) )

proof end;

theorem Th32: :: BORSUK_1:32

for XX being non empty TopSpace
for X being non empty SubSpace of XX
for D being non empty a_partition of the carrier of X
for x being Point of XX st not x in the carrier of X holds
{x} in TrivExt D

proof end;

theorem Th33: :: BORSUK_1:33

for XX being non empty TopSpace
for X being non empty SubSpace of XX
for D being non empty a_partition of the carrier of X
for W being Point of XX st W in the carrier of X holds
(Proj (TrivExt D)) . W = (Proj D) . W

proof end;

theorem Th34: :: BORSUK_1:34

for XX being non empty TopSpace
for X being non empty SubSpace of XX
for D being non empty a_partition of the carrier of X
for W being Point of XX st not W in the carrier of X holds
(Proj (TrivExt D)) . W = {W}

proof end;

theorem Th35: :: BORSUK_1:35

for XX being non empty TopSpace
for X being non empty SubSpace of XX
for D being non empty a_partition of the carrier of X
for W, W9 being Point of XX st not W in the carrier of X & (Proj (TrivExt D)) . W = (Proj (TrivExt D)) . W9 holds
W = W9

proof end;

theorem Th36: :: BORSUK_1:36

for XX being non empty TopSpace
for X being non empty SubSpace of XX
for D being non empty a_partition of the carrier of X
for e being Point of XX st (Proj (TrivExt D)) . e in the carrier of (space D) holds
e in the carrier of X

proof end;

theorem Th37: :: BORSUK_1:37

for XX being non empty TopSpace
for X being non empty SubSpace of XX
for D being non empty a_partition of the carrier of X
for e being set st e in the carrier of X holds
(Proj (TrivExt D)) . e in the carrier of (space D)

proof end;

::
:: Upper Semicontinuous Decompositions
::

definition

let X be non empty TopSpace;

mode u.s.c._decomposition of X -> non empty a_partition of the carrier of X means :Def10: :: BORSUK_1:def 10
for A being Subset of X st A in it holds
for V being a_neighborhood of A ex W being Subset of X st
( W is open & A c= W & W c= V & ( for B being Subset of X st B in it & B meets W holds
B c= W ) );

correctness
existence
ex b₁ being non empty a_partition of the carrier of X st
for A being Subset of X st A in b₁ holds
for V being a_neighborhood of A ex W being Subset of X st
( W is open & A c= W & W c= V & ( for B being Subset of X st B in b₁ & B meets W holds
B c= W ) );

proof end;

end;

:: deftheorem Def10 defines u.s.c._decomposition BORSUK_1:def 10 :
for X being non empty TopSpace
for b₂ being non empty a_partition of the carrier of X holds
( b₂ is u.s.c._decomposition of X iff for A being Subset of X st A in b₂ holds
for V being a_neighborhood of A ex W being Subset of X st
( W is open & A c= W & W c= V & ( for B being Subset of X st B in b₂ & B meets W holds
B c= W ) ) );

theorem Th38: :: BORSUK_1:38

for X being non empty TopSpace
for D being u.s.c._decomposition of X
for t being Point of (space D)
for G being a_neighborhood of (Proj D) " {t} holds (Proj D) .: G is a_neighborhood of t

proof end;

theorem Th39: :: BORSUK_1:39

for X being non empty TopSpace holds TrivDecomp X is u.s.c._decomposition of X

proof end;

definition

let X be TopSpace;
let IT be SubSpace of X;

attr IT is closed means :Def11: :: BORSUK_1:def 11
for A being Subset of X st A = the carrier of IT holds
A is closed ;

end;

:: deftheorem Def11 defines closed BORSUK_1:def 11 :
for X being TopSpace
for IT being SubSpace of X holds
( IT is closed iff for A being Subset of X st A = the carrier of IT holds
A is closed );

Lm2: for T being TopStruct holds TopStruct(# the carrier of T, the topology of T #) is SubSpace of T

proof end;

registration

let X be TopSpace;

cluster strict TopSpace-like closed for SubSpace of X;

existence
ex b₁ being SubSpace of X st
( b₁ is strict & b₁ is closed )

proof end;

end;

registration

let X be non empty TopSpace;

cluster non empty strict TopSpace-like closed for SubSpace of X;

existence
ex b₁ being SubSpace of X st
( b₁ is strict & b₁ is closed & not b₁ is empty )

proof end;

end;

definition

let XX be non empty TopSpace;
let X be non empty closed SubSpace of XX;
let D be u.s.c._decomposition of X;
:: original: TrivExt
redefine func TrivExt D -> u.s.c._decomposition of XX;
correctness
coherence
TrivExt D is u.s.c._decomposition of XX;

proof end;

end;

definition

let X be non empty TopSpace;
let IT be u.s.c._decomposition of X;

attr IT is DECOMPOSITION-like means :Def12: :: BORSUK_1:def 12
for A being Subset of X st A in IT holds
A is compact ;

end;

:: deftheorem Def12 defines DECOMPOSITION-like BORSUK_1:def 12 :
for X being non empty TopSpace
for IT being u.s.c._decomposition of X holds
( IT is DECOMPOSITION-like iff for A being Subset of X st A in IT holds
A is compact );

:: upper semicontinuous decomposition into compacta

registration

let X be non empty TopSpace;

cluster non empty with_non-empty_elements DECOMPOSITION-like for u.s.c._decomposition of X;

existence
ex b₁ being u.s.c._decomposition of X st b₁ is DECOMPOSITION-like

proof end;

end;

definition

let X be non empty TopSpace;
mode DECOMPOSITION of X is DECOMPOSITION-like u.s.c._decomposition of X;

end;

definition

let XX be non empty TopSpace;
let X be non empty closed SubSpace of XX;
let D be DECOMPOSITION of X;
:: original: TrivExt
redefine func TrivExt D -> DECOMPOSITION of XX;
correctness
coherence
TrivExt D is DECOMPOSITION of XX;

proof end;

end;

definition

let X be non empty TopSpace;
let Y be non empty closed SubSpace of X;
let D be DECOMPOSITION of Y;
:: original: space
redefine func space D -> strict closed SubSpace of space (TrivExt D);
correctness
coherence
space D is strict closed SubSpace of space (TrivExt D);

proof end;

end;

Lm3: TopSpaceMetr RealSpace = TopStruct(# the carrier of RealSpace,(Family_open_set RealSpace) #)
by PCOMPS_1:def 5;

definition

func I[01] -> TopStruct means :Def13: :: BORSUK_1:def 13
for P being Subset of (TopSpaceMetr RealSpace) st P = [.0,1.] holds
it = (TopSpaceMetr RealSpace) | P;

existence
ex b₁ being TopStruct st
for P being Subset of (TopSpaceMetr RealSpace) st P = [.0,1.] holds
b₁ = (TopSpaceMetr RealSpace) | P

proof end;

uniqueness
for b₁, b₂ being TopStruct st ( for P being Subset of (TopSpaceMetr RealSpace) st P = [.0,1.] holds
b₁ = (TopSpaceMetr RealSpace) | P ) & ( for P being Subset of (TopSpaceMetr RealSpace) st P = [.0,1.] holds
b₂ = (TopSpaceMetr RealSpace) | P ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def13 defines I[01] BORSUK_1:def 13 :
for b₁ being TopStruct holds
( b₁ = I[01] iff for P being Subset of (TopSpaceMetr RealSpace) st P = [.0,1.] holds
b₁ = (TopSpaceMetr RealSpace) | P );

registration

cluster I[01] -> non empty strict TopSpace-like ;

coherence
( I[01] is strict & not I[01] is empty & I[01] is TopSpace-like )

proof end;

end;

theorem Th40: :: BORSUK_1:40

the carrier of I[01] = [.0,1.]

proof end;

definition

func 0[01] -> Point of I[01] equals :: BORSUK_1:def 14
0 ;

coherence
0 is Point of I[01] by Th40, XXREAL_1:1;

func 1[01] -> Point of I[01] equals :: BORSUK_1:def 15
1;

coherence
1 is Point of I[01] by Th40, XXREAL_1:1;

end;

:: deftheorem defines 0[01] BORSUK_1:def 14 :
0[01] = 0 ;

:: deftheorem defines 1[01] BORSUK_1:def 15 :
1[01] = 1;

definition

let A be non empty TopSpace;
let B be non empty SubSpace of A;
let F be Function of A,B;

attr F is being_a_retraction means :: BORSUK_1:def 16
for W being Point of A st W in the carrier of B holds
F . W = W;

end;

:: deftheorem defines being_a_retraction BORSUK_1:def 16 :
for A being non empty TopSpace
for B being non empty SubSpace of A
for F being Function of A,B holds
( F is being_a_retraction iff for W being Point of A st W in the carrier of B holds
F . W = W );

definition

let X be non empty TopSpace;
let Y be non empty SubSpace of X;

pred Y is_a_retract_of X means :: BORSUK_1:def 17
ex F being continuous Function of X,Y st F is being_a_retraction ;

pred Y is_an_SDR_of X means :: BORSUK_1:def 18
ex H being continuous Function of [:X,I[01]:],X st
for A being Point of X holds
( H . [A,0[01]] = A & H . [A,1[01]] in the carrier of Y & ( A in the carrier of Y implies for T being Point of I[01] holds H . [A,T] = A ) );

end;

:: deftheorem defines is_a_retract_of BORSUK_1:def 17 :
for X being non empty TopSpace
for Y being non empty SubSpace of X holds
( Y is_a_retract_of X iff ex F being continuous Function of X,Y st F is being_a_retraction );

:: deftheorem defines is_an_SDR_of BORSUK_1:def 18 :
for X being non empty TopSpace
for Y being non empty SubSpace of X holds
( Y is_an_SDR_of X iff ex H being continuous Function of [:X,I[01]:],X st
for A being Point of X holds
( H . [A,0[01]] = A & H . [A,1[01]] in the carrier of Y & ( A in the carrier of Y implies for T being Point of I[01] holds H . [A,T] = A ) ) );

theorem :: BORSUK_1:41

for XX being non empty TopSpace
for X being non empty closed SubSpace of XX
for D being DECOMPOSITION of X st X is_a_retract_of XX holds
space D is_a_retract_of space (TrivExt D)

proof end;

:: Borsuk Theorem on Decomposition of Strong Deformation Retracts

theorem :: BORSUK_1:42

for XX being non empty TopSpace
for X being non empty closed SubSpace of XX
for D being DECOMPOSITION of X st X is_an_SDR_of XX holds
space D is_an_SDR_of space (TrivExt D)

proof end;

theorem :: BORSUK_1:43

for r being Real holds
( ( 0 <= r & r <= 1 ) iff r in the carrier of I[01] )

proof end;