Lm2: for GX being non empty 1-sorted
for R being Subset of GX
for p being Element of GX holds
( p in R ` iff not p in R )
by SUBSET_1:50, SUBSET_1:54;

theorem Th39: :: TOPS_1:39

for GX being non empty TopSpace
for R being Subset of GX
for p being Point of GX holds
( p in Cl R iff for T being Subset of GX st T is open & p in T holds
R meets T )

proof end;

theorem Th40: :: TOPS_1:40

for TS being TopSpace
for Q, K being Subset of TS st Q is open holds
Q /\ (Cl K) c= Cl (Q /\ K)

proof end;

theorem :: TOPS_1:41

for TS being TopSpace
for Q, K being Subset of TS st Q is open holds
Cl (Q /\ (Cl K)) = Cl (Q /\ K)

proof end;

definition

let GX be TopStruct ;
let R be Subset of GX;
func Int R -> Subset of GX equals :: TOPS_1:def 1
(Cl (R ` )) ` ;
coherence ;

end;

:: deftheorem defines Int TOPS_1:def 1 :

theorem :: TOPS_1:42

canceled;

theorem Th43: :: TOPS_1:43

for TS being TopSpace holds Int ([#] TS) = [#] TS

proof end;

theorem Th44: :: TOPS_1:44

for GX being TopStruct
for T being Subset of GX holds Int T c= T

proof end;

theorem Th45: :: TOPS_1:45

for GX being TopStruct
for T being Subset of GX holds Int (Int T) = Int T by Th26;

theorem Th46: :: TOPS_1:46

for TS being TopSpace
for K, L being Subset of TS holds (Int K) /\ (Int L) = Int (K /\ L)

proof end;

theorem Th47: :: TOPS_1:47

for GX being TopStruct holds Int ({} GX) = {} GX

proof end;

registration

let GX be TopStruct ;
cluster Int ({} GX) -> empty ;
coherence by Th47;

end;

theorem Th48: :: TOPS_1:48

for GX being TopStruct
for T, W being Subset of GX st T c= W holds
Int T c= Int W

proof end;

theorem Th49: :: TOPS_1:49

for GX being TopStruct
for T, W being Subset of GX holds (Int T) \/ (Int W) c= Int (T \/ W)

proof end;

theorem :: TOPS_1:50

for TS being TopSpace
for K, L being Subset of TS holds Int (K \ L) c= (Int K) \ (Int L)

proof end;

theorem :: TOPS_1:51

for TS being TopSpace
for K being Subset of TS holds Int K is open ;

registration

let T be TopSpace;
let K be Subset of T;
cluster Int K -> open ;
coherence ;

end;

theorem Th52: :: TOPS_1:52

for TS being TopSpace holds {} TS is open

proof end;

theorem Th53: :: TOPS_1:53

for TS being TopSpace holds [#] TS is open

proof end;

registration

let T be TopSpace;
cluster {} T -> open closed ;
coherence by Th52;
cluster [#] T -> open ;
coherence by Th53;

end;

registration

let TS be TopSpace;
cluster empty -> open closed Element of K10(the carrier of TS);
coherence

proof end;

end;

registration

let T be TopSpace;
cluster open closed Element of K10(the carrier of T);
existence

proof end;

end;

registration

let T be non empty TopSpace;
cluster non empty open closed Element of K10(the carrier of T);
existence

proof end;

end;

theorem Th54: :: TOPS_1:54

for TS being TopSpace
for x being set
for K being Subset of TS holds
( x in Int K iff ex Q being Subset of TS st
( Q is open & Q c= K & x in Q ) )

proof end;

theorem Th55: :: TOPS_1:55

for TS being TopSpace
for GX being TopStruct
for P being Subset of TS
for R being Subset of GX holds
( ( R is open implies Int R = R ) & ( Int P = P implies P is open ) )

proof end;

theorem :: TOPS_1:56

for GX being TopStruct
for S, T being Subset of GX st S is open & S c= T holds
S c= Int T

proof end;

theorem :: TOPS_1:57

for TS being TopSpace
for P being Subset of TS holds
( P is open iff for x being set holds
( x in P iff ex Q being Subset of TS st
( Q is open & Q c= P & x in Q ) ) )

proof end;

theorem Th58: :: TOPS_1:58

for GX being TopStruct
for T being Subset of GX holds Cl (Int T) = Cl (Int (Cl (Int T)))

proof end;

theorem :: TOPS_1:59

for GX being TopStruct
for R being Subset of GX st R is open holds
Cl (Int (Cl R)) = Cl R

proof end;

definition

let GX be TopStruct ;
let R be Subset of GX;
func Fr R -> Subset of GX equals :: TOPS_1:def 2
(Cl R) /\ (Cl (R ` ));
coherence ;

end;

:: deftheorem defines Fr TOPS_1:def 2 :

theorem :: TOPS_1:60

canceled;

theorem Th61: :: TOPS_1:61

for GX being non empty TopSpace
for R being Subset of GX
for p being Point of GX holds
( p in Fr R iff for S being Subset of GX st S is open & p in S holds
( R meets S & R ` meets S ) )

proof end;

theorem :: TOPS_1:62

for GX being TopStruct
for T being Subset of GX holds Fr T = Fr (T ` ) ;

theorem Th63: :: TOPS_1:63

for GX being TopStruct
for T being Subset of GX holds Fr T c= Cl T by XBOOLE_1:17;

theorem :: TOPS_1:64

for GX being TopStruct
for T being Subset of GX holds Fr T = ((Cl (T ` )) /\ T) \/ ((Cl T) \ T)

proof end;

theorem Th65: :: TOPS_1:65

for GX being TopStruct
for T being Subset of GX holds Cl T = T \/ (Fr T)

proof end;

theorem Th66: :: TOPS_1:66

for TS being TopSpace
for K, L being Subset of TS holds Fr (K /\ L) c= (Fr K) \/ (Fr L)

proof end;

theorem Th67: :: TOPS_1:67

for TS being TopSpace
for K, L being Subset of TS holds Fr (K \/ L) c= (Fr K) \/ (Fr L)

proof end;

theorem Th68: :: TOPS_1:68

for GX being TopStruct
for T being Subset of GX holds Fr (Fr T) c= Fr T

proof end;

theorem :: TOPS_1:69

for GX being TopStruct
for R being Subset of GX st R is closed holds
Fr R c= R

proof end;

Lm3: for TS being TopSpace
for K, L being Subset of TS holds Fr K c= ((Fr (K \/ L)) \/ (Fr (K /\ L))) \/ ((Fr K) /\ (Fr L))

proof end;

theorem :: TOPS_1:70

for TS being TopSpace
for K, L being Subset of TS holds (Fr K) \/ (Fr L) = ((Fr (K \/ L)) \/ (Fr (K /\ L))) \/ ((Fr K) /\ (Fr L))

proof end;

theorem :: TOPS_1:71

for GX being TopStruct
for T being Subset of GX holds Fr (Int T) c= Fr T

proof end;

theorem :: TOPS_1:72

for GX being TopStruct
for T being Subset of GX holds Fr (Cl T) c= Fr T

proof end;

theorem Th73: :: TOPS_1:73

for GX being TopStruct
for T being Subset of GX holds Int T misses Fr T

proof end;

theorem Th74: :: TOPS_1:74

for GX being TopStruct
for T being Subset of GX holds Int T = T \ (Fr T)

proof end;

Lm4: for GX being TopStruct
for T being Subset of GX holds Fr T = (Cl T) \ (Int T)

proof end;

Lm5: for TS being TopSpace
for K being Subset of TS holds Cl (Fr K) = Fr K

proof end;

Lm6: for TS being TopSpace
for K being Subset of TS holds Int (Fr (Fr K)) = {}

proof end;

theorem :: TOPS_1:75

for TS being TopSpace
for K being Subset of TS holds Fr (Fr (Fr K)) = Fr (Fr K)

proof end;

Lm7: for X, Y, Z being set st X c= Z & Y = Z \ X holds
X c= Z \ Y

proof end;

theorem Th76: :: TOPS_1:76

for TS being TopSpace
for P being Subset of TS holds
( P is open iff Fr P = (Cl P) \ P )

proof end;

theorem Th77: :: TOPS_1:77

for TS being TopSpace
for P being Subset of TS holds
( P is closed iff Fr P = P \ (Int P) )

proof end;

definition

let GX be TopStruct ;
let R be Subset of GX;
attr R is dense means :Def3: :: TOPS_1:def 3
Cl R = [#] GX;

end;

:: deftheorem Def3 defines dense TOPS_1:def 3 :

registration

let GX be TopStruct ;
cluster [#] GX -> dense ;
coherence

proof end;

end;

theorem :: TOPS_1:78

canceled;

theorem :: TOPS_1:79

for GX being TopStruct
for R, S being Subset of GX st R is dense & R c= S holds
S is dense

proof end;

theorem Th80: :: TOPS_1:80

for TS being TopSpace
for P being Subset of TS holds
( P is dense iff for Q being Subset of TS st Q <> {} & Q is open holds
P meets Q )

proof end;

theorem :: TOPS_1:81

for TS being TopSpace
for P being Subset of TS st P is dense holds
for Q being Subset of TS st Q is open holds
Cl Q = Cl (Q /\ P)

proof end;

theorem :: TOPS_1:82

for TS being TopSpace
for P, Q being Subset of TS st P is dense & Q is dense & Q is open holds
P /\ Q is dense

proof end;

definition

let GX be TopStruct ;
let R be Subset of GX;
attr R is boundary means :Def4: :: TOPS_1:def 4
R ` is dense;

end;

:: deftheorem Def4 defines boundary TOPS_1:def 4 :

registration

let GX be TopStruct ;
cluster empty -> boundary Element of K10(the carrier of GX);
coherence

proof end;

end;

registration

let GX be TopStruct ;
cluster empty boundary Element of K10(the carrier of GX);
existence

proof end;

end;

theorem :: TOPS_1:83

canceled;

theorem Th84: :: TOPS_1:84

for GX being TopStruct
for R being Subset of GX holds
( R is boundary iff Int R = {} )

proof end;

registration

let GX be TopStruct ;
let R be boundary Subset of GX;
cluster Int R -> empty boundary ;
coherence by Th84;

end;

theorem Th85: :: TOPS_1:85

for TS being TopSpace
for P, Q being Subset of TS st P is boundary & Q is boundary & Q is closed holds
P \/ Q is boundary

proof end;

theorem :: TOPS_1:86

for TS being TopSpace
for P being Subset of TS holds
( P is boundary iff for Q being Subset of TS st Q c= P & Q is open holds
Q = {} )

proof end;

theorem :: TOPS_1:87

for TS being TopSpace
for P being Subset of TS st P is closed holds
( P is boundary iff for Q being Subset of TS st Q <> {} & Q is open holds
ex G being Subset of TS st
( G c= Q & G <> {} & G is open & P misses G ) )

proof end;

theorem :: TOPS_1:88

for GX being TopStruct
for R being Subset of GX holds
( R is boundary iff R c= Fr R )

proof end;

definition

let GX be TopStruct ;
let R be Subset of GX;
attr R is nowhere_dense means :Def5: :: TOPS_1:def 5
Cl R is boundary;

end;

:: deftheorem Def5 defines nowhere_dense TOPS_1:def 5 :

registration

let TS be TopSpace;
cluster empty -> nowhere_dense Element of K10(the carrier of TS);
coherence

proof end;

end;

registration

let TS be TopSpace;
cluster empty open closed boundary nowhere_dense Element of K10(the carrier of TS);
existence

proof end;

end;

theorem :: TOPS_1:89

canceled;

theorem :: TOPS_1:90

for TS being TopSpace
for P, Q being Subset of TS st P is nowhere_dense & Q is nowhere_dense holds
P \/ Q is nowhere_dense

proof end;

theorem Th91: :: TOPS_1:91

for GX being TopStruct
for R being Subset of GX st R is nowhere_dense holds
R ` is dense

proof end;

registration

let TS be TopSpace;
let R be nowhere_dense Subset of TS;
cluster R ` -> dense ;
coherence by Th91;

end;

theorem Th92: :: TOPS_1:92

for GX being TopStruct
for R being Subset of GX st R is nowhere_dense holds
R is boundary

proof end;

registration

let TS be TopSpace;
cluster nowhere_dense -> boundary Element of K10(the carrier of TS);
coherence by Th92;

end;

theorem Th93: :: TOPS_1:93

for GX being TopStruct
for S being Subset of GX st S is boundary & S is closed holds
S is nowhere_dense

proof end;

registration

let TS be TopSpace;
cluster closed boundary -> boundary nowhere_dense Element of K10(the carrier of TS);
coherence by Th93;

end;

theorem :: TOPS_1:94

for GX being TopStruct
for R being Subset of GX st R is closed holds
( R is nowhere_dense iff R = Fr R )

proof end;

theorem Th95: :: TOPS_1:95

for TS being TopSpace
for P being Subset of TS st P is open holds
Fr P is nowhere_dense

proof end;

registration

let TS be TopSpace;
let P be open Subset of TS;
cluster Fr P -> boundary nowhere_dense ;
coherence by Th95;

end;

theorem Th96: :: TOPS_1:96

for TS being TopSpace
for P being Subset of TS st P is closed holds
Fr P is nowhere_dense

proof end;

registration

let TS be TopSpace;
let P be closed Subset of TS;
cluster Fr P -> boundary nowhere_dense ;
coherence by Th96;

end;

theorem Th97: :: TOPS_1:97

for TS being TopSpace
for P being Subset of TS st P is open & P is nowhere_dense holds
P = {}

proof end;

registration

let TS be TopSpace;
cluster open nowhere_dense -> empty open closed boundary nowhere_dense Element of K10(the carrier of TS);
coherence by Th97;

end;

definition

let GX be TopStruct ;
let R be Subset of GX;
attr R is condensed means :Def6: :: TOPS_1:def 6
( Int (Cl R) c= R & R c= Cl (Int R) );
attr R is closed_condensed means :Def7: :: TOPS_1:def 7
R = Cl (Int R);
attr R is open_condensed means :Def8: :: TOPS_1:def 8
R = Int (Cl R);

end;

:: deftheorem Def6 defines condensed TOPS_1:def 6 :

:: deftheorem Def7 defines closed_condensed TOPS_1:def 7 :

:: deftheorem Def8 defines open_condensed TOPS_1:def 8 :

theorem :: TOPS_1:98

canceled;

theorem :: TOPS_1:99

canceled;

theorem :: TOPS_1:100

canceled;

theorem Th101: :: TOPS_1:101

for GX being TopStruct
for R being Subset of GX holds
( R is open_condensed iff R ` is closed_condensed )

proof end;

theorem Th102: :: TOPS_1:102

for GX being TopStruct
for R being Subset of GX st R is closed_condensed holds
Fr (Int R) = Fr R

proof end;

theorem :: TOPS_1:103

for GX being TopStruct
for R being Subset of GX st R is closed_condensed holds
Fr R c= Cl (Int R)

proof end;

theorem Th104: :: TOPS_1:104

for GX being TopStruct
for R being Subset of GX st R is open_condensed holds
( Fr R = Fr (Cl R) & Fr (Cl R) = (Cl R) \ R )

proof end;

theorem :: TOPS_1:105

for GX being TopStruct
for R being Subset of GX st R is open & R is closed holds
( R is closed_condensed iff R is open_condensed )

proof end;

theorem Th106: :: TOPS_1:106

for TS being TopSpace
for GX being TopStruct
for P being Subset of TS
for R being Subset of GX holds
( ( R is closed & R is condensed implies R is closed_condensed ) & ( P is closed_condensed implies ( P is closed & P is condensed ) ) )

proof end;

theorem :: TOPS_1:107

for TS being TopSpace
for GX being TopStruct
for P being Subset of TS
for R being Subset of GX holds
( ( R is open & R is condensed implies R is open_condensed ) & ( P is open_condensed implies ( P is open & P is condensed ) ) )

proof end;

theorem Th108: :: TOPS_1:108

for TS being TopSpace
for P, Q being Subset of TS st P is closed_condensed & Q is closed_condensed holds
P \/ Q is closed_condensed

proof end;

theorem :: TOPS_1:109

for TS being TopSpace
for P, Q being Subset of TS st P is open_condensed & Q is open_condensed holds
P /\ Q is open_condensed

proof end;

theorem :: TOPS_1:110

for TS being TopSpace
for P being Subset of TS st P is condensed holds
Int (Fr P) = {}

proof end;

theorem :: TOPS_1:111

for GX being TopStruct
for R being Subset of GX st R is condensed holds
( Int R is condensed & Cl R is condensed )

proof end;