:: TEX_1 semantic presentation

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 )
;

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

proof end;

end;

:: deftheorem Def1 defines trivial TEX_1:def 1 :

registration

let A be non empty set ;
cluster 1-sorted(# A #) -> non empty ;
coherence by STRUCT_0:def 1;

end;

registration

cluster strict non empty trivial 1-sorted ;
existence

proof end;

cluster strict non empty non trivial 1-sorted ;
existence

proof end;

end;

registration

cluster non empty trivial strict TopStruct ;
existence

proof end;

cluster non empty non trivial strict TopStruct ;
existence

proof end;

end;

theorem :: TEX_1:8

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

proof end;

registration

cluster non empty trivial strict TopStruct ;
existence

proof end;

end;

registration

cluster non empty trivial -> non empty discrete anti-discrete TopStruct ;
coherence

proof end;

cluster non empty discrete anti-discrete -> non empty trivial TopStruct ;
coherence

proof end;

end;

registration

cluster non empty non trivial strict TopStruct ;
existence

proof end;

end;

registration

cluster non empty non discrete -> non empty non trivial TopStruct ;
coherence

proof end;

cluster non empty non anti-discrete -> non empty non trivial TopStruct ;
coherence

proof end;

end;

definition

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

proof end;

end;

:: deftheorem defines cobool TEX_1:def 2 :

registration

let D be set ;
cluster cobool D -> non empty ;
coherence ;

end;

definition

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

end;

:: deftheorem defines ADTS TEX_1:def 3 :

registration

let D be set ;
cluster ADTS D -> strict TopSpace-like anti-discrete ;
coherence

proof end;

end;

registration

let D be non empty set ;
cluster ADTS D -> non empty strict TopSpace-like anti-discrete ;
coherence ;

end;

theorem Th9: :: TEX_1:9

for X being non empty anti-discrete TopSpace holds TopStruct(# the carrier of X,the topology of X #) = ADTS the carrier of X by TDLAT_3:def 2;

theorem :: TEX_1:10

for X being non empty TopSpace st TopStruct(# the carrier of X,the topology of X #) = TopStruct(# the carrier of (ADTS the carrier of X),the topology of (ADTS the carrier of X) #) holds
X is anti-discrete by TDLAT_3:def 2;

theorem Th11: :: TEX_1:11

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 ) )

proof end;

theorem Th12: :: TEX_1:12

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 Th13: :: TEX_1:13

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 Th14: :: TEX_1:14

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_1:15

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_1:16

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_1:17

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

proof end;

end;

theorem Th18: :: TEX_1:18

for X being non empty discrete TopSpace holds TopStruct(# the carrier of X,the topology of X #) = 1TopSp the carrier of X

proof end;

theorem :: TEX_1:19

for X being non empty TopSpace st TopStruct(# the carrier of X,the topology of X #) = TopStruct(# the carrier of (1TopSp the carrier of X),the topology of (1TopSp the carrier of X) #) holds
X is discrete

proof end;

theorem :: TEX_1:20

for X being non empty discrete TopSpace
for A being Subset of X holds
( Cl A = A & Int A = A )

proof end;

theorem :: TEX_1:21

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_1:22

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

proof end;

theorem Th23: :: TEX_1:23

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 non trivial strict discrete non anti-discrete TopStruct ;
existence

proof end;

cluster non empty non trivial strict non discrete anti-discrete TopStruct ;
existence

proof end;

end;

definition

let D be set ;
let F be Subset-Family of D;
let S be set ;
:: original: \
redefine func F \ S -> Subset-Family of D;
coherence

proof end;

end;

definition

let D be set ;
let d0 be Element of D;
canceled;
func STS D,d0 -> TopStruct equals :: TEX_1:def 5
TopStruct(# D,((bool D) \ { A where A is Subset of D : ( d0 in A & A <> D ) } ) #);
coherence ;

end;

:: deftheorem TEX_1:def 4 :

:: deftheorem defines STS TEX_1:def 5 :

registration

let D be set ;
let d0 be Element of D;
cluster STS D,d0 -> strict TopSpace-like ;
coherence

proof end;

end;

registration

let D be non empty set ;
let d0 be Element of D;
cluster STS D,d0 -> non empty strict TopSpace-like ;
coherence ;

end;

theorem Th24: :: TEX_1:24

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 Th25: :: TEX_1:25

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 Th26: :: TEX_1:26

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_1:27

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 non trivial strict non discrete non anti-discrete TopStruct ;
existence

proof end;

end;

theorem Th28: :: TEX_1:28

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 Th29: :: TEX_1:29

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_1:30

proof end;

theorem :: TEX_1:31

proof end;

Lm2: now

let D be non empty set ; :: thesis:
let d0 be Element of D; :: thesis:
let G be set ; :: thesis:
assume A1: G = { P where P is Subset of D : ( d0 in P & P <> D ) } ; :: thesis:
assume A2: D = {d0} ; :: thesis:
assume A3: G <> {} ; :: thesis:
consider x being Element of G;
x in G by A3;
then consider S being Subset of D such that
x = S and
A4: d0 in S and
A5: S <> D by A1;

A6: now

assume D \ S = {} ; :: thesis:
then D c= S by XBOOLE_1:37;
hence contradiction by A5, XBOOLE_0:def 10; :: thesis:

end;

consider d1 being Element of D \ S;
reconsider d1 = d1 as Element of D by A6, XBOOLE_0:def 4;
d0 <> d1 by A4, A6, XBOOLE_0:def 4;
hence contradiction by A2, TARSKI:def 1; :: thesis:

end;

theorem :: TEX_1:32

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_1:33

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_1:34

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;

definition

let X be non empty TopSpace;
redefine attr X is discrete means :Def6: :: TEX_1:def 6
for A being non empty Subset of X holds not A is boundary;
compatibility

proof end;

end;

:: deftheorem Def6 defines discrete TEX_1:def 6 :

theorem :: TEX_1:35

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 non almost_discrete -> non empty non trivial non discrete non anti-discrete TopStruct ;
coherence

proof end;

end;

definition

let X be non empty TopSpace;
redefine attr X is almost_discrete means :Def7: :: TEX_1:def 7
for A being non empty Subset of X holds not A is nowhere_dense;
compatibility

proof end;

end;

:: deftheorem Def7 defines almost_discrete TEX_1:def 7 :

theorem Th36: :: TEX_1:36

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 Th37: :: TEX_1:37

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_1:38

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 non trivial strict non discrete non anti-discrete almost_discrete TopStruct ;
existence

proof end;

end;

theorem Th39: :: TEX_1:39

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 non trivial strict non discrete non anti-discrete non almost_discrete TopStruct ;
existence

proof end;

end;

theorem :: TEX_1:40

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_1:41

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;