:: On Discrete and Almost Discrete Topological Spaces
:: by Zbigniew Karno
::
:: Received April 6, 1993
:: Copyright (c) 1993-2021 Association of Mizar Users

theorem Th1: :: TEX_1:1

for X being non empty TopSpace
for D, B being Subset of X
for C being Subset of (X modified_with_respect_to D) st B = C & B is open holds
C is open

proof end;

theorem :: TEX_1:2

for X being non empty TopSpace
for D, B being Subset of X
for C being Subset of (X modified_with_respect_to D) st B = C & B is closed holds
C is closed

proof end;

theorem Th3: :: TEX_1:3

for X being non empty TopSpace
for D being Subset of X
for C being Subset of (X modified_with_respect_to (D `)) st C = D holds
C is closed

proof end;

theorem Th4: :: TEX_1:4

for X being non empty TopSpace
for D being Subset of X
for C being Subset of (X modified_with_respect_to D) st C = D & D is dense holds
( C is dense & C is open )

proof end;

theorem Th5: :: TEX_1:5

for X being non empty TopSpace
for D being Subset of X
for C being Subset of (X modified_with_respect_to D) st D c= C & D is dense holds
C is everywhere_dense

proof end;

theorem Th6: :: TEX_1:6

for X being non empty TopSpace
for D being Subset of X
for C being Subset of (X modified_with_respect_to (D `)) st C = D & D is boundary holds
( C is boundary & C is closed )

proof end;

theorem Th7: :: TEX_1:7

for X being non empty TopSpace
for D being Subset of X
for C being Subset of (X modified_with_respect_to (D `)) st C c= D & D is boundary holds
C is nowhere_dense

proof end;

Lm1: for X, Y being set holds
( X c= Y iff X is Subset of Y )
;

:: 2. Trivial Topological Spaces.

definition

let Y be non empty 1-sorted ;

redefine attr Y is trivial means :Def1: :: TEX_1:def 1
ex d being Element of Y st the carrier of Y = {d};

compatibility
( Y is trivial iff ex d being Element of Y st the carrier of Y = {d} )

proof end;

end;

:: deftheorem Def1 defines trivial TEX_1:def 1 :
for Y being non empty 1-sorted holds
( Y is trivial iff ex d being Element of Y st the carrier of Y = {d} );

registration

cluster 1 -element strict for TopStruct ;

existence
ex b₁ being TopStruct st
( b₁ is 1 -element & b₁ is strict )

proof end;

cluster non trivial strict for TopStruct ;

existence
ex b₁ being TopStruct st
( not b₁ is trivial & b₁ is strict )

proof end;

end;

theorem :: TEX_1:8

for Y being 1 -element TopStruct st not the topology of Y is empty & Y is almost_discrete holds
Y is TopSpace-like

proof end;

registration

cluster 1 -element strict TopSpace-like for TopStruct ;

existence
ex b₁ being TopSpace st
( b₁ is 1 -element & b₁ is strict )

proof end;

end;

registration

cluster 1 -element TopSpace-like -> 1 -element discrete anti-discrete for TopStruct ;

coherence
for b₁ being 1 -element TopSpace holds
( b₁ is anti-discrete & b₁ is discrete )

proof end;

cluster non empty TopSpace-like discrete anti-discrete -> non empty trivial for TopStruct ;

coherence
for b₁ being non empty TopSpace st b₁ is discrete & b₁ is anti-discrete holds
b₁ is trivial

proof end;

end;

registration

cluster non trivial strict TopSpace-like for TopStruct ;

existence
ex b₁ being TopSpace st
( not b₁ is trivial & b₁ is strict )

proof end;

end;

registration

cluster non empty TopSpace-like non discrete -> non empty non trivial for TopStruct ;

coherence
for b₁ being non empty TopSpace st not b₁ is discrete holds
not b₁ is trivial ;

cluster non empty TopSpace-like non anti-discrete -> non empty non trivial for TopStruct ;

coherence
for b₁ being non empty TopSpace st not b₁ is anti-discrete holds
not b₁ is trivial ;

end;

:: 3. Examples of Discrete and Anti-discrete Topological Spaces.

definition

let D be set ;

func cobool D -> Subset-Family of D equals :: TEX_1:def 2
{{},D};

coherence
{{},D} is Subset-Family of D

proof end;

end;

:: deftheorem defines cobool TEX_1:def 2 :
for D being set holds cobool D = {{},D};

registration

let D be set ;

cluster cobool D -> non empty ;

coherence
not cobool D is empty ;

end;

definition

let D be set ;

func ADTS D -> TopStruct equals :: TEX_1:def 3
TopStruct(# D,(cobool D) #);

coherence
TopStruct(# D,(cobool D) #) is TopStruct ;

end;

:: deftheorem defines ADTS TEX_1:def 3 :
for D being set holds ADTS D = TopStruct(# D,(cobool D) #);

registration

let D be set ;

cluster ADTS D -> strict TopSpace-like anti-discrete ;

coherence
( ADTS D is strict & ADTS D is anti-discrete & ADTS D is TopSpace-like )

proof end;

end;

registration

let D be non empty set ;

cluster ADTS D -> non empty ;

coherence
not ADTS D is empty ;

end;

theorem Th9: :: TEX_1:9

for X being non empty anti-discrete TopSpace
for A being Subset of X holds
( ( A is empty implies Cl A = {} ) & ( not A is empty implies Cl A = the carrier of X ) ) by PCOMPS_1:2, TDLAT_3:19;

theorem Th10: :: TEX_1:10

for X being non empty anti-discrete TopSpace
for A being Subset of X holds
( ( A <> the carrier of X implies Int A = {} ) & ( A = the carrier of X implies Int A = the carrier of X ) )

proof end;

theorem Th11: :: TEX_1:11

for X being non empty TopSpace st ( for A being Subset of X st not A is empty holds
Cl A = the carrier of X ) holds
X is anti-discrete

proof end;

theorem Th12: :: TEX_1:12

for X being non empty TopSpace st ( for A being Subset of X st A <> the carrier of X holds
Int A = {} ) holds
X is anti-discrete

proof end;

theorem :: TEX_1:13

for X being non empty anti-discrete TopSpace
for A being Subset of X holds
( ( A <> {} implies A is dense ) & ( A <> the carrier of X implies A is boundary ) )

proof end;

theorem :: TEX_1:14

for X being non empty TopSpace st ( for A being Subset of X st A <> {} holds
A is dense ) holds
X is anti-discrete

proof end;

theorem :: TEX_1:15

for X being non empty TopSpace st ( for A being Subset of X st A <> the carrier of X holds
A is boundary ) holds
X is anti-discrete

proof end;

registration

let D be set ;

cluster 1TopSp D -> discrete ;

coherence
1TopSp D is discrete ;

end;

theorem :: TEX_1:16

for X being non empty discrete TopSpace
for A being Subset of X holds
( Cl A = A & Int A = A ) by TDLAT_3:16, TDLAT_3:15, PRE_TOPC:22, TOPS_1:23;

theorem :: TEX_1:17

for X being non empty TopSpace st ( for A being Subset of X holds Cl A = A ) holds
X is discrete

proof end;

theorem :: TEX_1:18

for X being non empty TopSpace st ( for A being Subset of X holds Int A = A ) holds
X is discrete

proof end;

theorem Th19: :: TEX_1:19

for D being non empty set holds
( ADTS D = 1TopSp D iff ex d0 being Element of D st D = {d0} )

proof end;

registration

cluster non empty strict TopSpace-like discrete non anti-discrete for TopStruct ;

existence
ex b₁ being TopSpace st
( b₁ is discrete & not b₁ is anti-discrete & b₁ is strict & not b₁ is empty )

proof end;

cluster non empty strict TopSpace-like non discrete anti-discrete for TopStruct ;

existence
ex b₁ being TopSpace st
( b₁ is anti-discrete & not b₁ is discrete & b₁ is strict & not b₁ is empty )

proof end;

end;

:: 4. An Example of a Topological Space.

definition

let D be set ;
let d0 be Element of D;

func STS (D,d0) -> TopStruct equals :: TEX_1:def 4
TopStruct(# D,((bool D) \ { A where A is Subset of D : ( d0 in A & A <> D ) } ) #);

coherence
TopStruct(# D,((bool D) \ { A where A is Subset of D : ( d0 in A & A <> D ) } ) #) is TopStruct ;

end;

:: deftheorem defines STS TEX_1:def 4 :
for D being set
for d0 being Element of D holds STS (D,d0) = TopStruct(# D,((bool D) \ { A where A is Subset of D : ( d0 in A & A <> D ) } ) #);

registration

let D be set ;
let d0 be Element of D;

cluster STS (D,d0) -> strict TopSpace-like ;

coherence
( STS (D,d0) is strict & STS (D,d0) is TopSpace-like )

proof end;

end;

registration

let D be non empty set ;
let d0 be Element of D;

cluster STS (D,d0) -> non empty ;

coherence
not STS (D,d0) is empty ;

end;

theorem Th20: :: TEX_1:20

for D being non empty set
for d0 being Element of D
for A being Subset of (STS (D,d0)) holds
( ( {d0} c= A implies A is closed ) & ( not A is empty & A is closed implies {d0} c= A ) )

proof end;

theorem Th21: :: TEX_1:21

for D being non empty set
for d0 being Element of D st not D \ {d0} is empty holds
for A being Subset of (STS (D,d0)) holds
( ( A = {d0} implies ( A is closed & A is boundary ) ) & ( not A is empty & A is closed & A is boundary implies A = {d0} ) )

proof end;

theorem Th22: :: TEX_1:22

for D being non empty set
for d0 being Element of D
for A being Subset of (STS (D,d0)) holds
( ( A c= D \ {d0} implies A is open ) & ( A <> D & A is open implies A c= D \ {d0} ) )

proof end;

theorem :: TEX_1:23

for D being non empty set
for d0 being Element of D st not D \ {d0} is empty holds
for A being Subset of (STS (D,d0)) holds
( ( A = D \ {d0} implies ( A is open & A is dense ) ) & ( A <> D & A is open & A is dense implies A = D \ {d0} ) )

proof end;

registration

cluster non empty strict TopSpace-like non discrete non anti-discrete for TopStruct ;

existence
ex b₁ being TopSpace st
( not b₁ is anti-discrete & not b₁ is discrete & b₁ is strict & not b₁ is empty )

proof end;

end;

theorem Th24: :: TEX_1:24

for D being non empty set
for d0 being Element of D
for Y being non empty TopSpace holds
( TopStruct(# the carrier of Y, the topology of Y #) = TopStruct(# the carrier of (STS (D,d0)), the topology of (STS (D,d0)) #) iff ( the carrier of Y = D & ( for A being Subset of Y holds
( ( {d0} c= A implies A is closed ) & ( not A is empty & A is closed implies {d0} c= A ) ) ) ) )

proof end;

theorem Th25: :: TEX_1:25

for D being non empty set
for d0 being Element of D
for Y being non empty TopSpace holds
( TopStruct(# the carrier of Y, the topology of Y #) = TopStruct(# the carrier of (STS (D,d0)), the topology of (STS (D,d0)) #) iff ( the carrier of Y = D & ( for A being Subset of Y holds
( ( A c= D \ {d0} implies A is open ) & ( A <> D & A is open implies A c= D \ {d0} ) ) ) ) )

proof end;

theorem :: TEX_1:26

for D being non empty set
for d0 being Element of D
for Y being non empty TopSpace holds
( TopStruct(# the carrier of Y, the topology of Y #) = TopStruct(# the carrier of (STS (D,d0)), the topology of (STS (D,d0)) #) iff ( the carrier of Y = D & ( for A being non empty Subset of Y holds Cl A = A \/ {d0} ) ) )

proof end;

theorem :: TEX_1:27

for D being non empty set
for d0 being Element of D
for Y being non empty TopSpace holds
( TopStruct(# the carrier of Y, the topology of Y #) = TopStruct(# the carrier of (STS (D,d0)), the topology of (STS (D,d0)) #) iff ( the carrier of Y = D & ( for A being Subset of Y st A <> D holds
Int A = A \ {d0} ) ) )

proof end;

Lm2: now :: thesis: for D being non empty set
for d0 being Element of D
for G being set st G = { P where P is Subset of D : ( d0 in P & P <> D ) } & D = {d0} holds
not G <> {}

let D be non empty set ; :: thesis: for d0 being Element of D
for G being set st G = { P where P is Subset of D : ( d0 in P & P <> D ) } & D = {d0} holds
not G <> {}
let d0 be Element of D; :: thesis: for G being set st G = { P where P is Subset of D : ( d0 in P & P <> D ) } & D = {d0} holds
not G <> {}
let G be set ; :: thesis: ( G = { P where P is Subset of D : ( d0 in P & P <> D ) } & D = {d0} implies not G <> {} )
assume A1: G = { P where P is Subset of D : ( d0 in P & P <> D ) } ; :: thesis: ( D = {d0} implies not G <> {} )
set x = the Element of G;
assume A2: D = {d0} ; :: thesis: not G <> {}
assume G <> {} ; :: thesis: contradiction
then the Element of G in G ;
then consider S being Subset of D such that
the Element of G = S and
A3: d0 in S and
A4: S <> D by A1;
set d1 = the Element of D \ S;

A5: now :: thesis: not D \ S = {}

assume D \ S = {} ; :: thesis: contradiction
then D c= S by XBOOLE_1:37;
hence contradiction by A4, XBOOLE_0:def 10; :: thesis: verum

end;

then reconsider d1 = the Element of D \ S as Element of D by XBOOLE_0:def 5;
d0 <> d1 by A3, A5, XBOOLE_0:def 5;
hence contradiction by A2, TARSKI:def 1; :: thesis: verum

end;

theorem :: TEX_1:28

for D being non empty set
for d0 being Element of D holds
( STS (D,d0) = ADTS D iff D = {d0} )

proof end;

theorem :: TEX_1:29

for D being non empty set
for d0 being Element of D holds
( STS (D,d0) = 1TopSp D iff D = {d0} )

proof end;

theorem :: TEX_1:30

for D being non empty set
for d0 being Element of D
for A being Subset of (STS (D,d0)) st A = {d0} holds
1TopSp D = (STS (D,d0)) modified_with_respect_to A

proof end;

:: 5. Discrete and Almost Discrete Spaces.

definition

let X be non empty TopSpace;

redefine attr X is discrete means :: TEX_1:def 5
for A being non empty Subset of X holds not A is boundary ;

compatibility
( X is discrete iff for A being non empty Subset of X holds not A is boundary )

proof end;

end;

:: deftheorem defines discrete TEX_1:def 5 :
for X being non empty TopSpace holds
( X is discrete iff for A being non empty Subset of X holds not A is boundary );

theorem :: TEX_1:31

for X being non empty TopSpace holds
( X is discrete iff for A being Subset of X st A <> the carrier of X holds
not A is dense )

proof end;

registration

cluster non empty TopSpace-like non almost_discrete -> non empty non discrete non anti-discrete for TopStruct ;

coherence
for b₁ being non empty TopSpace st not b₁ is almost_discrete holds
( not b₁ is discrete & not b₁ is anti-discrete ) ;

end;

definition

let X be non empty TopSpace;

redefine attr X is almost_discrete means :: TEX_1:def 6
for A being non empty Subset of X holds not A is nowhere_dense ;

compatibility
( X is almost_discrete iff for A being non empty Subset of X holds not A is nowhere_dense )

proof end;

end;

:: deftheorem defines almost_discrete TEX_1:def 6 :
for X being non empty TopSpace holds
( X is almost_discrete iff for A being non empty Subset of X holds not A is nowhere_dense );

theorem Th32: :: TEX_1:32

for X being non empty TopSpace holds
( X is almost_discrete iff for A being Subset of X st A <> the carrier of X holds
not A is everywhere_dense )

proof end;

theorem Th33: :: TEX_1:33

for X being non empty TopSpace holds
( not X is almost_discrete iff ex A being non empty Subset of X st
( A is boundary & A is closed ) )

proof end;

theorem :: TEX_1:34

for X being non empty TopSpace holds
( not X is almost_discrete iff ex A being Subset of X st
( A <> the carrier of X & A is dense & A is open ) )

proof end;

registration

cluster non empty strict TopSpace-like non discrete non anti-discrete almost_discrete for TopStruct ;

existence
ex b₁ being TopSpace st
( b₁ is almost_discrete & not b₁ is discrete & not b₁ is anti-discrete & b₁ is strict & not b₁ is empty )

proof end;

end;

theorem Th35: :: TEX_1:35

for C being non empty set
for c0 being Element of C holds
( not C \ {c0} is empty iff not STS (C,c0) is almost_discrete )

proof end;

registration

cluster non empty strict TopSpace-like non almost_discrete for TopStruct ;

existence
ex b₁ being TopSpace st
( not b₁ is almost_discrete & b₁ is strict & not b₁ is empty )

proof end;

end;

theorem :: TEX_1:36

for X being non empty TopSpace
for A being non empty Subset of X st A is boundary holds
not X modified_with_respect_to (A `) is almost_discrete

proof end;

theorem :: TEX_1:37

for X being non empty TopSpace
for A being Subset of X st A <> the carrier of X & A is dense holds
not X modified_with_respect_to A is almost_discrete

proof end;