:: by Miros{\l}aw Wysocki and Agata Darmochwa\l

::

:: Received April 28, 1989

:: Copyright (c) 1990-2012 Association of Mizar Users

begin

theorem :: TOPS_1:1

registration
end;

registration
end;

theorem Th7: :: TOPS_1:7

for GX being TopStruct

for R, S being Subset of GX st R is closed & S is closed holds

Cl (R /\ S) = (Cl R) /\ (Cl S)

for R, S being Subset of GX st R is closed & S is closed holds

Cl (R /\ S) = (Cl R) /\ (Cl S)

proof end;

registration

let TS be TopSpace;

let P, Q be closed Subset of TS;

coherence

for b_{1} being Subset of TS st b_{1} = P /\ Q holds

b_{1} is closed

for b_{1} being Subset of TS st b_{1} = P \/ Q holds

b_{1} is closed

end;
let P, Q be closed Subset of TS;

coherence

for b

b

proof end;

coherence for b

b

proof end;

theorem :: TOPS_1:8

theorem :: TOPS_1:9

registration

let TS be TopSpace;

let P, Q be open Subset of TS;

coherence

for b_{1} being Subset of TS st b_{1} = P /\ Q holds

b_{1} is open

for b_{1} being Subset of TS st b_{1} = P \/ Q holds

b_{1} is open

end;
let P, Q be open Subset of TS;

coherence

for b

b

proof end;

coherence for b

b

proof end;

theorem :: TOPS_1:10

theorem :: TOPS_1:11

theorem Th12: :: TOPS_1:12

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 )

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;

definition

let GX be TopStruct ;

let R be Subset of GX;

coherence

(Cl (R `)) ` is Subset of GX ;

projectivity

for b_{1}, b_{2} being Subset of GX st b_{1} = (Cl (b_{2} `)) ` holds

b_{1} = (Cl (b_{1} `)) `
;

end;
let R be Subset of GX;

coherence

(Cl (R `)) ` is Subset of GX ;

projectivity

for b

b

:: deftheorem defines Int TOPS_1:def 1 :

for GX being TopStruct

for R being Subset of GX holds Int R = (Cl (R `)) ` ;

for GX being TopStruct

for R being Subset of GX holds Int R = (Cl (R `)) ` ;

registration
end;

registration
end;

registration

let T be TopSpace;

coherence

for b_{1} being Subset of T st b_{1} is empty holds

b_{1} is open

[#] T is open

end;
coherence

for b

b

proof end;

coherence [#] T is open

proof end;

registration

let T be TopSpace;

existence

ex b_{1} being Subset of T st

( b_{1} is open & b_{1} is closed )

end;
existence

ex b

( b

proof end;

registration

let T be non empty TopSpace;

existence

ex b_{1} being Subset of T st

( not b_{1} is empty & b_{1} is open & b_{1} is closed )

end;
existence

ex b

( not b

proof end;

theorem Th22: :: TOPS_1:22

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 ) )

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 Th23: :: TOPS_1:23

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 ) )

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_1:25

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 ) ) )

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;

::

:: FRONTIER OF A SET

::

:: FRONTIER OF A SET

::

definition
end;

:: deftheorem defines Fr TOPS_1:def 2 :

for GX being TopStruct

for R being Subset of GX holds Fr R = (Cl R) /\ (Cl (R `));

for GX being TopStruct

for R being Subset of GX holds Fr R = (Cl R) /\ (Cl (R `));

registration
end;

theorem Th28: :: TOPS_1:28

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 ) )

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;

Lm1: 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_1:36

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))

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

proof end;

Lm2: for GX being TopStruct

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

proof end;

Lm3: for TS being TopSpace

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

proof end;

Lm4: for TS being TopSpace

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

proof end;

Lm5: for X, Y, Z being set st X c= Z & Y = Z \ X holds

X c= Z \ Y

proof end;

::

:: DENSE, BOUNDARY AND NOWHEREDENSE SETS

::

:: DENSE, BOUNDARY AND NOWHEREDENSE SETS

::

:: deftheorem Def3 defines dense TOPS_1:def 3 :

for GX being TopStruct

for R being Subset of GX holds

( R is dense iff Cl R = [#] GX );

for GX being TopStruct

for R being Subset of GX holds

( R is dense iff Cl R = [#] GX );

registration

let GX be TopStruct ;

coherence

[#] GX is dense

ex b_{1} being Subset of GX st b_{1} is dense

end;
coherence

[#] GX is dense

proof end;

existence ex b

proof end;

theorem Th45: :: TOPS_1:45

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 )

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 Th46: :: TOPS_1:46

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)

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_1:47

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

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

P /\ Q is dense

proof end;

:: deftheorem Def4 defines boundary TOPS_1:def 4 :

for GX being TopStruct

for R being Subset of GX holds

( R is boundary iff R ` is dense );

for GX being TopStruct

for R being Subset of GX holds

( R is boundary iff R ` is dense );

registration

let GX be TopStruct ;

coherence

for b_{1} being Subset of GX st b_{1} is empty holds

b_{1} is boundary

end;
coherence

for b

b

proof end;

registration
end;

registration
end;

theorem Th49: :: TOPS_1:49

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

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_1:50

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 = {} )

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_1:51

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 ) )

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;

registration

let GX be non empty TopSpace;

coherence

not [#] GX is boundary

ex b_{1} being Subset of GX st

( not b_{1} is boundary & not b_{1} is empty )

end;
coherence

not [#] GX is boundary

proof end;

existence ex b

( not b

proof end;

:: deftheorem Def5 defines nowhere_dense TOPS_1:def 5 :

for GX being TopStruct

for R being Subset of GX holds

( R is nowhere_dense iff Cl R is boundary );

for GX being TopStruct

for R being Subset of GX holds

( R is nowhere_dense iff Cl R is boundary );

registration

let TS be TopSpace;

coherence

for b_{1} being Subset of TS st b_{1} is empty holds

b_{1} is nowhere_dense

end;
coherence

for b

b

proof end;

registration
end;

theorem :: TOPS_1:53

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

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

P \/ Q is nowhere_dense

proof end;

registration
end;

registration

let TS be TopSpace;

coherence

for b_{1} being Subset of TS st b_{1} is nowhere_dense holds

b_{1} is boundary
by Th55;

end;
coherence

for b

b

theorem Th56: :: TOPS_1:56

for GX being TopStruct

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

S is nowhere_dense

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;

coherence

for b_{1} being Subset of TS st b_{1} is boundary & b_{1} is closed holds

b_{1} is nowhere_dense
by Th56;

end;
coherence

for b

b

theorem :: TOPS_1:57

for GX being TopStruct

for R being Subset of GX st R is closed holds

( R is nowhere_dense iff R = Fr R )

for R being Subset of GX st R is closed holds

( R is nowhere_dense iff R = Fr R )

proof end;

registration
end;

registration
end;

registration

let TS be TopSpace;

coherence

for b_{1} being Subset of TS st b_{1} is open & b_{1} is nowhere_dense holds

b_{1} is empty
by Th60;

end;
coherence

for b

b

::

:: CLOSED AND OPEN DOMAIN, DOMAIN

::

:: CLOSED AND OPEN DOMAIN, DOMAIN

::

:: deftheorem Def6 defines condensed TOPS_1:def 6 :

for GX being TopStruct

for R being Subset of GX holds

( R is condensed iff ( Int (Cl R) c= R & R c= Cl (Int R) ) );

for GX being TopStruct

for R being Subset of GX holds

( R is condensed iff ( Int (Cl R) c= R & R c= Cl (Int R) ) );

:: deftheorem Def7 defines closed_condensed TOPS_1:def 7 :

for GX being TopStruct

for R being Subset of GX holds

( R is closed_condensed iff R = Cl (Int R) );

for GX being TopStruct

for R being Subset of GX holds

( R is closed_condensed iff R = Cl (Int R) );

:: deftheorem Def8 defines open_condensed TOPS_1:def 8 :

for GX being TopStruct

for R being Subset of GX holds

( R is open_condensed iff R = Int (Cl R) );

for GX being TopStruct

for R being Subset of GX holds

( R is open_condensed iff R = Int (Cl R) );

theorem Th61: :: TOPS_1:61

for GX being TopStruct

for R being Subset of GX holds

( R is open_condensed iff R ` is closed_condensed )

for R being Subset of GX holds

( R is open_condensed iff R ` is closed_condensed )

proof end;

theorem Th64: :: TOPS_1:64

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 )

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_1:65

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 )

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 Th66: :: TOPS_1:66

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 ) ) )

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_1:67

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 ) ) )

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 Th68: :: TOPS_1:68

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

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_1:69

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

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_1:71

for GX being TopStruct

for R being Subset of GX st R is condensed holds

( Int R is condensed & Cl R is condensed )

for R being Subset of GX st R is condensed holds

( Int R is condensed & Cl R is condensed )

proof end;

:: INTERIOR OF A SET

::