:: Maximal Anti-Discrete Subspaces of Topological Spaces
:: by Zbigniew Karno
::
:: Received July 26, 1994
:: Copyright (c) 1994-2021 Association of Mizar Users

:: 1. Properties of the Closure and the Interior Operations.

registration

let X be non empty TopSpace;
let A be non empty Subset of X;

cluster Cl A -> non empty ;

coherence
not Cl A is empty by PCOMPS_1:2;

end;

registration

let X be TopSpace;
let A be empty Subset of X;

cluster Cl A -> empty ;

coherence
Cl A is empty by PRE_TOPC:22;

end;

registration

let X be non empty TopSpace;
let A be non proper Subset of X;

cluster Cl A -> non proper ;

coherence
not Cl A is proper

proof end;

end;

registration

let X be non trivial TopSpace;
let A be non trivial Subset of X;

cluster Cl A -> non trivial ;

coherence
not Cl A is trivial

proof end;

end;

theorem Th1: :: TEX_4:1

for X being non empty TopSpace
for A being Subset of X holds Cl A = meet { F where F is Subset of X : ( F is closed & A c= F ) }

proof end;

theorem Th2: :: TEX_4:2

for X being non empty TopSpace
for x being Point of X holds Cl {x} = meet { F where F is Subset of X : ( F is closed & x in F ) }

proof end;

registration

let X be non empty TopSpace;
let A be non proper Subset of X;

cluster Int A -> non proper ;

coherence
not Int A is proper

proof end;

end;

registration

let X be non empty TopSpace;
let A be proper Subset of X;

cluster Int A -> proper ;

coherence
Int A is proper

proof end;

end;

registration

let X be non empty TopSpace;
let A be empty Subset of X;

cluster Int A -> empty ;

coherence
Int A is empty ;

end;

theorem :: TEX_4:3

for X being non empty TopSpace
for A being Subset of X holds Int A = union { G where G is Subset of X : ( G is open & G c= A ) }

proof end;

:: 2. Anti-discrete Subsets of Topological Structures.

definition

let Y be TopStruct ;
let IT be Subset of Y;

attr IT is anti-discrete means :: TEX_4:def 1
for x being Point of Y
for G being Subset of Y st G is open & x in G & x in IT holds
IT c= G;

end;

:: deftheorem defines anti-discrete TEX_4:def 1 :
for Y being TopStruct
for IT being Subset of Y holds
( IT is anti-discrete iff for x being Point of Y
for G being Subset of Y st G is open & x in G & x in IT holds
IT c= G );

definition

let Y be non empty TopStruct ;
let A be Subset of Y;

redefine attr A is anti-discrete means :: TEX_4:def 2
for x being Point of Y
for F being Subset of Y st F is closed & x in F & x in A holds
A c= F;

compatibility
( A is anti-discrete iff for x being Point of Y
for F being Subset of Y st F is closed & x in F & x in A holds
A c= F )

proof end;

end;

:: deftheorem defines anti-discrete TEX_4:def 2 :
for Y being non empty TopStruct
for A being Subset of Y holds
( A is anti-discrete iff for x being Point of Y
for F being Subset of Y st F is closed & x in F & x in A holds
A c= F );

definition

let Y be TopStruct ;
let A be Subset of Y;

redefine attr A is anti-discrete means :: TEX_4:def 3
for G being Subset of Y holds
( not G is open or A misses G or A c= G );

compatibility
( A is anti-discrete iff for G being Subset of Y holds
( not G is open or A misses G or A c= G ) )

proof end;

end;

:: deftheorem defines anti-discrete TEX_4:def 3 :
for Y being TopStruct
for A being Subset of Y holds
( A is anti-discrete iff for G being Subset of Y holds
( not G is open or A misses G or A c= G ) );

definition

let Y be TopStruct ;
let A be Subset of Y;

redefine attr A is anti-discrete means :: TEX_4:def 4
for F being Subset of Y holds
( not F is closed or A misses F or A c= F );

compatibility
( A is anti-discrete iff for F being Subset of Y holds
( not F is closed or A misses F or A c= F ) )

proof end;

end;

:: deftheorem defines anti-discrete TEX_4:def 4 :
for Y being TopStruct
for A being Subset of Y holds
( A is anti-discrete iff for F being Subset of Y holds
( not F is closed or A misses F or A c= F ) );

theorem Th4: :: TEX_4:4

for Y0, Y1 being TopStruct
for D0 being Subset of Y0
for D1 being Subset of Y1 st TopStruct(# the carrier of Y0, the topology of Y0 #) = TopStruct(# the carrier of Y1, the topology of Y1 #) & D0 = D1 & D0 is anti-discrete holds
D1 is anti-discrete

proof end;

theorem Th5: :: TEX_4:5

for Y being non empty TopStruct
for A, B being Subset of Y st B c= A & A is anti-discrete holds
B is anti-discrete

proof end;

theorem Th6: :: TEX_4:6

for Y being non empty TopStruct
for x being Point of Y holds {x} is anti-discrete

proof end;

theorem :: TEX_4:7

for Y being non empty TopStruct
for A being empty Subset of Y holds A is anti-discrete ;

definition

let Y be TopStruct ;
let IT be Subset-Family of Y;

attr IT is anti-discrete-set-family means :: TEX_4:def 5
for A being Subset of Y st A in IT holds
A is anti-discrete ;

end;

:: deftheorem defines anti-discrete-set-family TEX_4:def 5 :
for Y being TopStruct
for IT being Subset-Family of Y holds
( IT is anti-discrete-set-family iff for A being Subset of Y st A in IT holds
A is anti-discrete );

theorem Th8: :: TEX_4:8

for Y being non empty TopStruct
for F being Subset-Family of Y st F is anti-discrete-set-family & meet F <> {} holds
union F is anti-discrete

proof end;

theorem :: TEX_4:9

for Y being non empty TopStruct
for F being Subset-Family of Y st F is anti-discrete-set-family holds
meet F is anti-discrete

proof end;

definition

let Y be TopStruct ;
let x be Point of Y;

func MaxADSF x -> Subset-Family of Y equals :: TEX_4:def 6
{ A where A is Subset of Y : ( A is anti-discrete & x in A ) } ;

coherence
{ A where A is Subset of Y : ( A is anti-discrete & x in A ) } is Subset-Family of Y

proof end;

end;

:: deftheorem defines MaxADSF TEX_4:def 6 :
for Y being TopStruct
for x being Point of Y holds MaxADSF x = { A where A is Subset of Y : ( A is anti-discrete & x in A ) } ;

registration

let Y be non empty TopStruct ;
let x be Point of Y;

cluster MaxADSF x -> non empty ;

coherence
not MaxADSF x is empty

proof end;

end;

theorem Th10: :: TEX_4:10

for Y being non empty TopStruct
for x being Point of Y holds MaxADSF x is anti-discrete-set-family

proof end;

theorem Th11: :: TEX_4:11

for Y being non empty TopStruct
for x being Point of Y holds {x} = meet (MaxADSF x)

proof end;

theorem Th12: :: TEX_4:12

for Y being non empty TopStruct
for x being Point of Y holds {x} c= union (MaxADSF x)

proof end;

theorem Th13: :: TEX_4:13

for Y being non empty TopStruct
for x being Point of Y holds union (MaxADSF x) is anti-discrete

proof end;

:: 3. Maximal Anti-discrete Subsets of Topological Structures.

definition

let Y be TopStruct ;
let IT be Subset of Y;

attr IT is maximal_anti-discrete means :: TEX_4:def 7
( IT is anti-discrete & ( for D being Subset of Y st D is anti-discrete & IT c= D holds
IT = D ) );

end;

:: deftheorem defines maximal_anti-discrete TEX_4:def 7 :
for Y being TopStruct
for IT being Subset of Y holds
( IT is maximal_anti-discrete iff ( IT is anti-discrete & ( for D being Subset of Y st D is anti-discrete & IT c= D holds
IT = D ) ) );

theorem :: TEX_4:14

for Y0, Y1 being TopStruct
for D0 being Subset of Y0
for D1 being Subset of Y1 st TopStruct(# the carrier of Y0, the topology of Y0 #) = TopStruct(# the carrier of Y1, the topology of Y1 #) & D0 = D1 & D0 is maximal_anti-discrete holds
D1 is maximal_anti-discrete

proof end;

theorem Th15: :: TEX_4:15

for Y being non empty TopStruct
for A being empty Subset of Y holds not A is maximal_anti-discrete

proof end;

theorem Th16: :: TEX_4:16

for Y being non empty TopStruct
for A being non empty Subset of Y st A is anti-discrete & A is open holds
A is maximal_anti-discrete

proof end;

theorem Th17: :: TEX_4:17

for Y being non empty TopStruct
for A being non empty Subset of Y st A is anti-discrete & A is closed holds
A is maximal_anti-discrete

proof end;

definition

let Y be TopStruct ;
let x be Point of Y;

func MaxADSet x -> Subset of Y equals :: TEX_4:def 8
union (MaxADSF x);

coherence
union (MaxADSF x) is Subset of Y ;

end;

:: deftheorem defines MaxADSet TEX_4:def 8 :
for Y being TopStruct
for x being Point of Y holds MaxADSet x = union (MaxADSF x);

registration

let Y be non empty TopStruct ;
let x be Point of Y;

cluster MaxADSet x -> non empty ;

coherence
not MaxADSet x is empty

proof end;

end;

theorem :: TEX_4:18

for Y being non empty TopStruct
for x being Point of Y holds {x} c= MaxADSet x by Th12;

theorem Th19: :: TEX_4:19

for Y being non empty TopStruct
for D being Subset of Y
for x being Point of Y st D is anti-discrete & x in D holds
D c= MaxADSet x

proof end;

theorem Th20: :: TEX_4:20

for Y being non empty TopStruct
for x being Point of Y holds MaxADSet x is maximal_anti-discrete

proof end;

theorem Th21: :: TEX_4:21

for Y being non empty TopStruct
for x, y being Point of Y holds
( y in MaxADSet x iff MaxADSet y = MaxADSet x )

proof end;

theorem Th22: :: TEX_4:22

for Y being non empty TopStruct
for x, y being Point of Y holds
( MaxADSet x misses MaxADSet y or MaxADSet x = MaxADSet y )

proof end;

theorem Th23: :: TEX_4:23

for Y being non empty TopStruct
for F being Subset of Y
for x being Point of Y st F is closed & x in F holds
MaxADSet x c= F

proof end;

theorem Th24: :: TEX_4:24

for Y being non empty TopStruct
for G being Subset of Y
for x being Point of Y st G is open & x in G holds
MaxADSet x c= G

proof end;

theorem :: TEX_4:25

for Y being non empty TopSpace
for x being Point of Y holds MaxADSet x c= meet { F where F is Subset of Y : ( F is closed & x in F ) }

proof end;

theorem :: TEX_4:26

for Y being non empty TopSpace
for x being Point of Y holds MaxADSet x c= meet { G where G is Subset of Y : ( G is open & x in G ) }

proof end;

definition

let Y be non empty TopStruct ;
let A be Subset of Y;

redefine attr A is maximal_anti-discrete means :: TEX_4:def 9
ex x being Point of Y st
( x in A & A = MaxADSet x );

compatibility
( A is maximal_anti-discrete iff ex x being Point of Y st
( x in A & A = MaxADSet x ) )

proof end;

end;

:: deftheorem defines maximal_anti-discrete TEX_4:def 9 :
for Y being non empty TopStruct
for A being Subset of Y holds
( A is maximal_anti-discrete iff ex x being Point of Y st
( x in A & A = MaxADSet x ) );

theorem Th27: :: TEX_4:27

for Y being non empty TopStruct
for A being Subset of Y
for x being Point of Y st x in A & A is maximal_anti-discrete holds
A = MaxADSet x by Th21;

definition

let Y be non empty TopStruct ;
let A be non empty Subset of Y;

redefine attr A is maximal_anti-discrete means :: TEX_4:def 10
for x being Point of Y st x in A holds
A = MaxADSet x;

compatibility
( A is maximal_anti-discrete iff for x being Point of Y st x in A holds
A = MaxADSet x )

proof end;

end;

:: deftheorem defines maximal_anti-discrete TEX_4:def 10 :
for Y being non empty TopStruct
for A being non empty Subset of Y holds
( A is maximal_anti-discrete iff for x being Point of Y st x in A holds
A = MaxADSet x );

definition

let Y be non empty TopStruct ;
let A be Subset of Y;

func MaxADSet A -> Subset of Y equals :: TEX_4:def 11
union { (MaxADSet a) where a is Point of Y : a in A } ;

coherence
union { (MaxADSet a) where a is Point of Y : a in A } is Subset of Y

proof end;

end;

:: deftheorem defines MaxADSet TEX_4:def 11 :
for Y being non empty TopStruct
for A being Subset of Y holds MaxADSet A = union { (MaxADSet a) where a is Point of Y : a in A } ;

theorem Th28: :: TEX_4:28

for Y being non empty TopStruct
for x being Point of Y holds MaxADSet x = MaxADSet {x}

proof end;

theorem Th29: :: TEX_4:29

for Y being non empty TopStruct
for A being Subset of Y
for x being Point of Y st MaxADSet x meets MaxADSet A holds
MaxADSet x meets A

proof end;

theorem Th30: :: TEX_4:30

for Y being non empty TopStruct
for A being Subset of Y
for x being Point of Y st MaxADSet x meets MaxADSet A holds
MaxADSet x c= MaxADSet A

proof end;

theorem Th31: :: TEX_4:31

for Y being non empty TopStruct
for A, B being Subset of Y st A c= B holds
MaxADSet A c= MaxADSet B

proof end;

theorem Th32: :: TEX_4:32

for Y being non empty TopStruct
for A being Subset of Y holds A c= MaxADSet A

proof end;

theorem Th33: :: TEX_4:33

for Y being non empty TopStruct
for A being Subset of Y holds MaxADSet A = MaxADSet (MaxADSet A)

proof end;

theorem Th34: :: TEX_4:34

for Y being non empty TopStruct
for A, B being Subset of Y st A c= MaxADSet B holds
MaxADSet A c= MaxADSet B

proof end;

theorem Th35: :: TEX_4:35

for Y being non empty TopStruct
for A, B being Subset of Y st B c= MaxADSet A & A c= MaxADSet B holds
MaxADSet A = MaxADSet B by Th34;

theorem :: TEX_4:36

for Y being non empty TopStruct
for A, B being Subset of Y holds MaxADSet (A \/ B) = (MaxADSet A) \/ (MaxADSet B)

proof end;

theorem Th37: :: TEX_4:37

for Y being non empty TopStruct
for A, B being Subset of Y holds MaxADSet (A /\ B) c= (MaxADSet A) /\ (MaxADSet B)

proof end;

registration

let Y be non empty TopStruct ;
let A be non empty Subset of Y;

cluster MaxADSet A -> non empty ;

coherence
not MaxADSet A is empty by Th32, XBOOLE_1:3;

end;

registration

let Y be non empty TopStruct ;
let A be empty Subset of Y;

cluster MaxADSet A -> empty ;

coherence
MaxADSet A is empty

proof end;

end;

registration

let Y be non empty TopStruct ;
let A be non proper Subset of Y;

cluster MaxADSet A -> non proper ;

coherence
not MaxADSet A is proper

proof end;

end;

registration

let Y be non trivial TopStruct ;
let A be non trivial Subset of Y;

cluster MaxADSet A -> non trivial ;

coherence
not MaxADSet A is trivial

proof end;

end;

theorem Th38: :: TEX_4:38

for Y being non empty TopStruct
for G, A being Subset of Y st G is open & A c= G holds
MaxADSet A c= G

proof end;

theorem Th39: :: TEX_4:39

for Y being non empty TopSpace
for A being Subset of Y holds MaxADSet A c= meet { G where G is Subset of Y : ( G is open & A c= G ) }

proof end;

theorem Th40: :: TEX_4:40

for Y being non empty TopStruct
for F, A being Subset of Y st F is closed & A c= F holds
MaxADSet A c= F

proof end;

theorem Th41: :: TEX_4:41

for Y being non empty TopSpace
for A being Subset of Y holds MaxADSet A c= meet { F where F is Subset of Y : ( F is closed & A c= F ) }

proof end;

:: 4. Anti-discrete and Maximal Anti-discrete Subsets of Topological Spaces.

definition

let X be non empty TopSpace;
let A be Subset of X;

redefine attr A is anti-discrete means :: TEX_4:def 12
for x being Point of X st x in A holds
A c= Cl {x};

compatibility
( A is anti-discrete iff for x being Point of X st x in A holds
A c= Cl {x} )

proof end;

end;

:: deftheorem defines anti-discrete TEX_4:def 12 :
for X being non empty TopSpace
for A being Subset of X holds
( A is anti-discrete iff for x being Point of X st x in A holds
A c= Cl {x} );

definition

let X be non empty TopSpace;
let A be Subset of X;

redefine attr A is anti-discrete means :: TEX_4:def 13
for x being Point of X st x in A holds
Cl A = Cl {x};

compatibility
( A is anti-discrete iff for x being Point of X st x in A holds
Cl A = Cl {x} )

proof end;

end;

:: deftheorem defines anti-discrete TEX_4:def 13 :
for X being non empty TopSpace
for A being Subset of X holds
( A is anti-discrete iff for x being Point of X st x in A holds
Cl A = Cl {x} );

definition

let X be non empty TopSpace;
let A be Subset of X;

redefine attr A is anti-discrete means :: TEX_4:def 14
for x, y being Point of X st x in A & y in A holds
Cl {x} = Cl {y};

compatibility
( A is anti-discrete iff for x, y being Point of X st x in A & y in A holds
Cl {x} = Cl {y} )

proof end;

end;

:: deftheorem defines anti-discrete TEX_4:def 14 :
for X being non empty TopSpace
for A being Subset of X holds
( A is anti-discrete iff for x, y being Point of X st x in A & y in A holds
Cl {x} = Cl {y} );

theorem :: TEX_4:42

for X being non empty TopSpace
for x being Point of X
for D being Subset of X st D is anti-discrete & Cl {x} c= D holds
D = Cl {x}

proof end;

theorem :: TEX_4:43

for X being non empty TopSpace
for A being Subset of X holds
( ( A is anti-discrete & A is closed ) iff for x being Point of X st x in A holds
A = Cl {x} )

proof end;

theorem :: TEX_4:44

for X being non empty TopSpace
for A being Subset of X st A is anti-discrete & not A is open holds
A is boundary

proof end;

theorem Th45: :: TEX_4:45

for X being non empty TopSpace
for x being Point of X st Cl {x} = {x} holds
{x} is maximal_anti-discrete

proof end;

theorem Th46: :: TEX_4:46

for X being non empty TopSpace
for x being Point of X holds MaxADSet x c= meet { G where G is Subset of X : ( G is open & x in G ) }

proof end;

theorem Th47: :: TEX_4:47

for X being non empty TopSpace
for x being Point of X holds MaxADSet x c= meet { F where F is Subset of X : ( F is closed & x in F ) }

proof end;

theorem Th48: :: TEX_4:48

for X being non empty TopSpace
for x being Point of X holds MaxADSet x c= Cl {x}

proof end;

Lm1: for X being non empty TopSpace
for x, y being Point of X st MaxADSet x = MaxADSet y holds
Cl {x} = Cl {y}

proof end;

theorem Th49: :: TEX_4:49

for X being non empty TopSpace
for x, y being Point of X holds
( MaxADSet x = MaxADSet y iff Cl {x} = Cl {y} )

proof end;

theorem :: TEX_4:50

for X being non empty TopSpace
for x, y being Point of X holds
( MaxADSet x misses MaxADSet y iff Cl {x} <> Cl {y} ) by Th49, Th22;

definition

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

:: original: MaxADSet
redefine func MaxADSet x -> non empty Subset of X equals :: TEX_4:def 15
(Cl {x}) /\ (meet { G where G is Subset of X : ( G is open & x in G ) } );

compatibility
for b₁ being non empty Subset of X holds
( b₁ = MaxADSet x iff b₁ = (Cl {x}) /\ (meet { G where G is Subset of X : ( G is open & x in G ) } ) )

proof end;

coherence
MaxADSet x is non empty Subset of X ;

end;

:: deftheorem defines MaxADSet TEX_4:def 15 :
for X being non empty TopSpace
for x being Point of X holds MaxADSet x = (Cl {x}) /\ (meet { G where G is Subset of X : ( G is open & x in G ) } );

theorem Th51: :: TEX_4:51

for X being non empty TopSpace
for x, y being Point of X holds
( Cl {x} c= Cl {y} iff meet { G where G is Subset of X : ( G is open & y in G ) } c= meet { G where G is Subset of X : ( G is open & x in G ) } )

proof end;

theorem :: TEX_4:52

for X being non empty TopSpace
for x, y being Point of X holds
( Cl {x} c= Cl {y} iff MaxADSet y c= meet { G where G is Subset of X : ( G is open & x in G ) } )

proof end;

theorem Th53: :: TEX_4:53

for X being non empty TopSpace
for x, y being Point of X holds
( MaxADSet x misses MaxADSet y iff ( ex V being Subset of X st
( V is open & MaxADSet x c= V & V misses MaxADSet y ) or ex W being Subset of X st
( W is open & W misses MaxADSet x & MaxADSet y c= W ) ) )

proof end;

theorem :: TEX_4:54

for X being non empty TopSpace
for x, y being Point of X holds
( MaxADSet x misses MaxADSet y iff ( ex E being Subset of X st
( E is closed & MaxADSet x c= E & E misses MaxADSet y ) or ex F being Subset of X st
( F is closed & F misses MaxADSet x & MaxADSet y c= F ) ) )

proof end;

theorem Th55: :: TEX_4:55

for X being non empty TopSpace
for A being Subset of X holds MaxADSet A c= meet { G where G is Subset of X : ( G is open & A c= G ) } by Th39;

theorem Th56: :: TEX_4:56

for X being non empty TopSpace
for P being Subset of X st P is open holds
MaxADSet P = P

proof end;

theorem :: TEX_4:57

for X being non empty TopSpace
for A being Subset of X holds MaxADSet (Int A) = Int A by Th56;

theorem Th58: :: TEX_4:58

for X being non empty TopSpace
for A being Subset of X holds MaxADSet A c= meet { F where F is Subset of X : ( F is closed & A c= F ) } by Th41;

theorem Th59: :: TEX_4:59

for X being non empty TopSpace
for A being Subset of X holds MaxADSet A c= Cl A

proof end;

theorem Th60: :: TEX_4:60

for X being non empty TopSpace
for P being Subset of X st P is closed holds
MaxADSet P = P

proof end;

theorem :: TEX_4:61

for X being non empty TopSpace
for A being Subset of X holds MaxADSet (Cl A) = Cl A by Th60;

theorem :: TEX_4:62

for X being non empty TopSpace
for A being Subset of X holds Cl (MaxADSet A) = Cl A by Th59, TOPS_1:5, Th32, PRE_TOPC:19;

theorem :: TEX_4:63

for X being non empty TopSpace
for A, B being Subset of X st MaxADSet A = MaxADSet B holds
Cl A = Cl B

proof end;

theorem :: TEX_4:64

for X being non empty TopSpace
for P, Q being Subset of X st ( P is closed or Q is closed ) holds
MaxADSet (P /\ Q) = (MaxADSet P) /\ (MaxADSet Q)

proof end;

theorem :: TEX_4:65

for X being non empty TopSpace
for P, Q being Subset of X st ( P is open or Q is open ) holds
MaxADSet (P /\ Q) = (MaxADSet P) /\ (MaxADSet Q)

proof end;

theorem Th66: :: TEX_4:66

for Y being non empty TopStruct
for Y0 being SubSpace of Y
for A being Subset of Y st A = the carrier of Y0 & Y0 is anti-discrete holds
A is anti-discrete

proof end;

theorem Th67: :: TEX_4:67

for Y being non empty TopStruct
for Y0 being SubSpace of Y st Y0 is TopSpace-like holds
for A being Subset of Y st A = the carrier of Y0 & A is anti-discrete holds
Y0 is anti-discrete

proof end;

theorem :: TEX_4:68

for X being non empty TopSpace
for Y0 being non empty SubSpace of X st ( for X0 being open SubSpace of X holds
( Y0 misses X0 or Y0 is SubSpace of X0 ) ) holds
Y0 is anti-discrete

proof end;

theorem :: TEX_4:69

for X being non empty TopSpace
for Y0 being non empty SubSpace of X st ( for X0 being closed SubSpace of X holds
( Y0 misses X0 or Y0 is SubSpace of X0 ) ) holds
Y0 is anti-discrete

proof end;

theorem :: TEX_4:70

for X being non empty TopSpace
for Y0 being anti-discrete SubSpace of X
for X0 being open SubSpace of X holds
( Y0 misses X0 or Y0 is SubSpace of X0 )

proof end;

theorem :: TEX_4:71

for X being non empty TopSpace
for Y0 being anti-discrete SubSpace of X
for X0 being closed SubSpace of X holds
( Y0 misses X0 or Y0 is SubSpace of X0 )

proof end;

definition

let Y be non empty TopStruct ;
let IT be SubSpace of Y;

attr IT is maximal_anti-discrete means :: TEX_4:def 16
( IT is anti-discrete & ( for Y0 being SubSpace of Y st Y0 is anti-discrete & the carrier of IT c= the carrier of Y0 holds
the carrier of IT = the carrier of Y0 ) );

end;

:: deftheorem defines maximal_anti-discrete TEX_4:def 16 :
for Y being non empty TopStruct
for IT being SubSpace of Y holds
( IT is maximal_anti-discrete iff ( IT is anti-discrete & ( for Y0 being SubSpace of Y st Y0 is anti-discrete & the carrier of IT c= the carrier of Y0 holds
the carrier of IT = the carrier of Y0 ) ) );

registration

let Y be non empty TopStruct ;

cluster maximal_anti-discrete -> anti-discrete for SubSpace of Y;

coherence
for b₁ being SubSpace of Y st b₁ is maximal_anti-discrete holds
b₁ is anti-discrete ;

cluster non anti-discrete -> non maximal_anti-discrete for SubSpace of Y;

coherence
for b₁ being SubSpace of Y st not b₁ is anti-discrete holds
not b₁ is maximal_anti-discrete ;

end;

theorem Th72: :: TEX_4:72

for X being non empty TopSpace
for Y0 being non empty SubSpace of X
for A being Subset of X st A = the carrier of Y0 holds
( Y0 is maximal_anti-discrete iff A is maximal_anti-discrete )

proof end;

registration

let X be non empty TopSpace;

cluster non empty open anti-discrete -> non empty maximal_anti-discrete for SubSpace of X;

coherence
for b₁ being non empty SubSpace of X st b₁ is open & b₁ is anti-discrete holds
b₁ is maximal_anti-discrete

proof end;

cluster non empty open non maximal_anti-discrete -> non empty non anti-discrete for SubSpace of X;

coherence
for b₁ being non empty SubSpace of X st b₁ is open & not b₁ is maximal_anti-discrete holds
not b₁ is anti-discrete ;

cluster non empty anti-discrete non maximal_anti-discrete -> non empty non open for SubSpace of X;

coherence
for b₁ being non empty SubSpace of X st b₁ is anti-discrete & not b₁ is maximal_anti-discrete holds
not b₁ is open ;

cluster non empty closed anti-discrete -> non empty maximal_anti-discrete for SubSpace of X;

coherence
for b₁ being non empty SubSpace of X st b₁ is closed & b₁ is anti-discrete holds
b₁ is maximal_anti-discrete

proof end;

cluster non empty closed non maximal_anti-discrete -> non empty non anti-discrete for SubSpace of X;

coherence
for b₁ being non empty SubSpace of X st b₁ is closed & not b₁ is maximal_anti-discrete holds
not b₁ is anti-discrete ;

cluster non empty anti-discrete non maximal_anti-discrete -> non empty non closed for SubSpace of X;

coherence
for b₁ being non empty SubSpace of X st b₁ is anti-discrete & not b₁ is maximal_anti-discrete holds
not b₁ is closed ;

end;

definition

let Y be TopStruct ;
let x be Point of Y;

func MaxADSspace x -> strict SubSpace of Y means :Def17: :: TEX_4:def 17
the carrier of it = MaxADSet x;

existence
ex b₁ being strict SubSpace of Y st the carrier of b₁ = MaxADSet x

proof end;

uniqueness
for b₁, b₂ being strict SubSpace of Y st the carrier of b₁ = MaxADSet x & the carrier of b₂ = MaxADSet x holds
b₁ = b₂

proof end;

end;

:: deftheorem Def17 defines MaxADSspace TEX_4:def 17 :
for Y being TopStruct
for x being Point of Y
for b₃ being strict SubSpace of Y holds
( b₃ = MaxADSspace x iff the carrier of b₃ = MaxADSet x );

registration

let Y be non empty TopStruct ;
let x be Point of Y;

cluster MaxADSspace x -> non empty strict ;

coherence
not MaxADSspace x is empty

proof end;

end;

Lm2: for Y being non empty TopStruct
for X1, X2 being SubSpace of Y st the carrier of X1 c= the carrier of X2 holds
X1 is SubSpace of X2

proof end;

theorem :: TEX_4:73

for Y being non empty TopStruct
for x being Point of Y holds Sspace x is SubSpace of MaxADSspace x

proof end;

theorem :: TEX_4:74

for Y being non empty TopStruct
for x, y being Point of Y holds
( y is Point of (MaxADSspace x) iff TopStruct(# the carrier of (MaxADSspace y), the topology of (MaxADSspace y) #) = TopStruct(# the carrier of (MaxADSspace x), the topology of (MaxADSspace x) #) )

proof end;

theorem :: TEX_4:75

for Y being non empty TopStruct
for x, y being Point of Y holds
( the carrier of (MaxADSspace x) misses the carrier of (MaxADSspace y) or TopStruct(# the carrier of (MaxADSspace x), the topology of (MaxADSspace x) #) = TopStruct(# the carrier of (MaxADSspace y), the topology of (MaxADSspace y) #) )

proof end;

registration

let X be non empty TopSpace;

cluster strict TopSpace-like maximal_anti-discrete for SubSpace of X;

existence
ex b₁ being SubSpace of X st
( b₁ is maximal_anti-discrete & b₁ is strict )

proof end;

end;

registration

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

cluster MaxADSspace x -> strict maximal_anti-discrete ;

coherence
MaxADSspace x is maximal_anti-discrete

proof end;

end;

theorem :: TEX_4:76

for X being non empty TopSpace
for X0 being non empty closed SubSpace of X
for x being Point of X st x is Point of X0 holds
MaxADSspace x is SubSpace of X0

proof end;

theorem :: TEX_4:77

for X being non empty TopSpace
for X0 being non empty open SubSpace of X
for x being Point of X st x is Point of X0 holds
MaxADSspace x is SubSpace of X0

proof end;

theorem :: TEX_4:78

for X being non empty TopSpace
for x being Point of X st Cl {x} = {x} holds
Sspace x is maximal_anti-discrete

proof end;

notation

let Y be TopStruct ;
let A be Subset of Y;
synonym Sspace A for Y | A;

end;

Lm3: for Y being TopStruct
for A being Subset of Y holds the carrier of (Y | A) = A

proof end;

theorem :: TEX_4:79

for Y being non empty TopStruct
for A being non empty Subset of Y holds A is Subset of (Sspace A)

proof end;

theorem :: TEX_4:80

for Y being non empty TopStruct
for Y0 being SubSpace of Y
for A being non empty Subset of Y st A is Subset of Y0 holds
Sspace A is SubSpace of Y0

proof end;

registration

let Y be non trivial TopStruct ;

cluster strict non proper for SubSpace of Y;

existence
ex b₁ being SubSpace of Y st
( not b₁ is proper & b₁ is strict )

proof end;

end;

registration

let Y be non trivial TopStruct ;
let A be non trivial Subset of Y;

cluster Sspace A -> non trivial ;

coherence
not Sspace A is trivial by Lm3;

end;

registration

let Y be non empty TopStruct ;
let A be non proper Subset of Y;

cluster Sspace A -> non proper ;

coherence
not Sspace A is proper

proof end;

end;

definition

let Y be non empty TopStruct ;
let A be Subset of Y;

func MaxADSspace A -> strict SubSpace of Y means :Def18: :: TEX_4:def 18
the carrier of it = MaxADSet A;

existence
ex b₁ being strict SubSpace of Y st the carrier of b₁ = MaxADSet A

proof end;

uniqueness
for b₁, b₂ being strict SubSpace of Y st the carrier of b₁ = MaxADSet A & the carrier of b₂ = MaxADSet A holds
b₁ = b₂

proof end;

end;

:: deftheorem Def18 defines MaxADSspace TEX_4:def 18 :
for Y being non empty TopStruct
for A being Subset of Y
for b₃ being strict SubSpace of Y holds
( b₃ = MaxADSspace A iff the carrier of b₃ = MaxADSet A );

registration

let Y be non empty TopStruct ;
let A be non empty Subset of Y;

cluster MaxADSspace A -> non empty strict ;

coherence
not MaxADSspace A is empty

proof end;

end;

theorem :: TEX_4:81

for Y being non empty TopStruct
for A being non empty Subset of Y holds A is Subset of (MaxADSspace A)

proof end;

theorem :: TEX_4:82

for Y being non empty TopStruct
for A being non empty Subset of Y holds Sspace A is SubSpace of MaxADSspace A

proof end;

theorem :: TEX_4:83

for Y being non empty TopStruct
for x being Point of Y holds TopStruct(# the carrier of (MaxADSspace x), the topology of (MaxADSspace x) #) = TopStruct(# the carrier of (MaxADSspace {x}), the topology of (MaxADSspace {x}) #)

proof end;

theorem :: TEX_4:84

for Y being non empty TopStruct
for A, B being non empty Subset of Y st A c= B holds
MaxADSspace A is SubSpace of MaxADSspace B

proof end;

theorem :: TEX_4:85

for Y being non empty TopStruct
for A being non empty Subset of Y holds TopStruct(# the carrier of (MaxADSspace A), the topology of (MaxADSspace A) #) = TopStruct(# the carrier of (MaxADSspace (MaxADSet A)), the topology of (MaxADSspace (MaxADSet A)) #)

proof end;

theorem :: TEX_4:86

for Y being non empty TopStruct
for A, B being non empty Subset of Y st A is Subset of (MaxADSspace B) holds
MaxADSspace A is SubSpace of MaxADSspace B

proof end;

theorem :: TEX_4:87

for Y being non empty TopStruct
for A, B being non empty Subset of Y holds
( ( B is Subset of (MaxADSspace A) & A is Subset of (MaxADSspace B) ) iff TopStruct(# the carrier of (MaxADSspace A), the topology of (MaxADSspace A) #) = TopStruct(# the carrier of (MaxADSspace B), the topology of (MaxADSspace B) #) )

proof end;

registration

let Y be non trivial TopStruct ;
let A be non trivial Subset of Y;

cluster MaxADSspace A -> non trivial strict ;

coherence
not MaxADSspace A is trivial

proof end;

end;

registration

let Y be non empty TopStruct ;
let A be non proper Subset of Y;

cluster MaxADSspace A -> strict non proper ;

coherence
not MaxADSspace A is proper

proof end;

end;

theorem :: TEX_4:88

for X being non empty TopSpace
for X0 being open SubSpace of X
for A being non empty Subset of X st A is Subset of X0 holds
MaxADSspace A is SubSpace of X0

proof end;

theorem :: TEX_4:89

for X being non empty TopSpace
for X0 being closed SubSpace of X
for A being non empty Subset of X st A is Subset of X0 holds
MaxADSspace A is SubSpace of X0

proof end;