:: COMPTS_1 semantic presentation

definition
let T be 1-sorted ;
let F be Subset-Family of T;
let P be Subset of T;
pred c2 is_a_cover_of c3 means :Def1: :: COMPTS_1:def 1
P c= union F;
end;

:: deftheorem Def1 defines is_a_cover_of COMPTS_1:def 1 :
for T being 1-sorted
for F being Subset-Family of T
for P being Subset of T holds
( F is_a_cover_of P iff P c= union F );

definition
let F be set ;
attr a1 is centered means :Def2: :: COMPTS_1:def 2
( F <> {} & ( for G being set st G <> {} & G c= F & G is finite holds
meet G <> {} ) );
end;

:: deftheorem Def2 defines centered COMPTS_1:def 2 :
for F being set holds
( F is centered iff ( F <> {} & ( for G being set st G <> {} & G c= F & G is finite holds
meet G <> {} ) ) );

definition
let T be TopStruct ;
attr a1 is compact means :Def3: :: COMPTS_1:def 3
for F being Subset-Family of T st F is_a_cover_of T & F is open holds
ex G being Subset-Family of T st
( G c= F & G is_a_cover_of T & G is finite );
attr a1 is being_T2 means :Def4: :: COMPTS_1:def 4
for p, q being Point of T st p <> q holds
ex W, V being Subset of T st
( W is open & V is open & p in W & q in V & W misses V );
attr a1 is being_T3 means :: COMPTS_1:def 5
for p being Point of T
for P being Subset of T st P <> {} & P is closed & p in P ` holds
ex W, V being Subset of T st
( W is open & V is open & p in W & P c= V & W misses V );
attr a1 is being_T4 means :: COMPTS_1:def 6
for W, V being Subset of T st W <> {} & V <> {} & W is closed & V is closed & W misses V holds
ex P, Q being Subset of T st
( P is open & Q is open & W c= P & V c= Q & P misses Q );
end;

:: deftheorem Def3 defines compact COMPTS_1:def 3 :
for T being TopStruct holds
( T is compact iff for F being Subset-Family of T st F is_a_cover_of T & F is open holds
ex G being Subset-Family of T st
( G c= F & G is_a_cover_of T & G is finite ) );

:: deftheorem Def4 defines being_T2 COMPTS_1:def 4 :
for T being TopStruct holds
( T is being_T2 iff for p, q being Point of T st p <> q holds
ex W, V being Subset of T st
( W is open & V is open & p in W & q in V & W misses V ) );

:: deftheorem Def5 defines being_T3 COMPTS_1:def 5 :
for T being TopStruct holds
( T is being_T3 iff for p being Point of T
for P being Subset of T st P <> {} & P is closed & p in P ` holds
ex W, V being Subset of T st
( W is open & V is open & p in W & P c= V & W misses V ) );

:: deftheorem Def6 defines being_T4 COMPTS_1:def 6 :
for T being TopStruct holds
( T is being_T4 iff for W, V being Subset of T st W <> {} & V <> {} & W is closed & V is closed & W misses V holds
ex P, Q being Subset of T st
( P is open & Q is open & W c= P & V c= Q & P misses Q ) );

notation
let T be TopStruct ;
synonym Hausdorff c1 for being_T2 c1;
synonym c1 is_T2 for being_T2 c1;
synonym c1 is_T3 for being_T3 c1;
synonym c1 is_T4 for being_T4 c1;
end;

definition
let T be TopStruct ;
let P be Subset of T;
attr a2 is compact means :Def7: :: COMPTS_1:def 7
for F being Subset-Family of T st F is_a_cover_of P & F is open holds
ex G being Subset-Family of T st
( G c= F & G is_a_cover_of P & G is finite );
end;

:: deftheorem Def7 defines compact COMPTS_1:def 7 :
for T being TopStruct
for P being Subset of T holds
( P is compact iff for F being Subset-Family of T st F is_a_cover_of P & F is open holds
ex G being Subset-Family of T st
( G c= F & G is_a_cover_of P & G is finite ) );

theorem Th1: :: COMPTS_1:1
canceled;

theorem Th2: :: COMPTS_1:2
canceled;

theorem Th3: :: COMPTS_1:3
canceled;

theorem Th4: :: COMPTS_1:4
canceled;

theorem Th5: :: COMPTS_1:5
canceled;

theorem Th6: :: COMPTS_1:6
canceled;

theorem Th7: :: COMPTS_1:7
canceled;

theorem Th8: :: COMPTS_1:8
canceled;

theorem Th9: :: COMPTS_1:9
for T being TopStruct holds {} T is compact
proof end;

theorem Th10: :: COMPTS_1:10
for T being TopStruct holds
( T is compact iff [#] T is compact )
proof end;

theorem Th11: :: COMPTS_1:11
for T being TopStruct
for A being SubSpace of T
for Q being Subset of T st Q c= [#] A holds
( Q is compact iff for P being Subset of A st P = Q holds
P is compact )
proof end;

theorem Th12: :: COMPTS_1:12
for T being TopStruct
for P being Subset of T holds
( ( P = {} implies ( P is compact iff T | P is compact ) ) & ( T is TopSpace-like & P <> {} implies ( P is compact iff T | P is compact ) ) )
proof end;

theorem Th13: :: COMPTS_1:13
for T being non empty TopSpace holds
( T is compact iff for F being Subset-Family of T st F is centered & F is closed holds
meet F <> {} )
proof end;

theorem Th14: :: COMPTS_1:14
for T being non empty TopSpace holds
( T is compact iff for F being Subset-Family of T st F <> {} & F is closed & meet F = {} holds
ex G being Subset-Family of T st
( G <> {} & G c= F & G is finite & meet G = {} ) )
proof end;

theorem Th15: :: COMPTS_1:15
for TS being TopSpace st TS is_T2 holds
for A being Subset of TS st A <> {} & A is compact holds
for p being Point of TS st p in A ` holds
ex PS, QS being Subset of TS st
( PS is open & QS is open & p in PS & A c= QS & PS misses QS )
proof end;

theorem Th16: :: COMPTS_1:16
for TS being TopSpace
for PS being Subset of TS st TS is_T2 & PS is compact holds
PS is closed
proof end;

theorem Th17: :: COMPTS_1:17
for T being TopStruct
for P being Subset of T st T is compact & P is closed holds
P is compact
proof end;

theorem Th18: :: COMPTS_1:18
for TS being TopSpace
for PS, QS being Subset of TS st PS is compact & QS c= PS & QS is closed holds
QS is compact
proof end;

theorem Th19: :: COMPTS_1:19
for T being TopStruct
for P, Q being Subset of T st P is compact & Q is compact holds
P \/ Q is compact
proof end;

theorem Th20: :: COMPTS_1:20
for TS being TopSpace
for PS, QS being Subset of TS st TS is_T2 & PS is compact & QS is compact holds
PS /\ QS is compact
proof end;

theorem Th21: :: COMPTS_1:21
for TS being TopSpace st TS is_T2 & TS is compact holds
TS is_T3
proof end;

theorem Th22: :: COMPTS_1:22
for TS being TopSpace st TS is_T2 & TS is compact holds
TS is_T4
proof end;

theorem Th23: :: COMPTS_1:23
for T being TopStruct
for S being non empty TopStruct
for f being Function of T,S st T is compact & f is continuous & rng f = [#] S holds
S is compact
proof end;

theorem Th24: :: COMPTS_1:24
for T being TopStruct
for P being Subset of T
for S being non empty TopStruct
for f being Function of T,S st f is continuous & rng f = [#] S & P is compact holds
f .: P is compact
proof end;

theorem Th25: :: COMPTS_1:25
for TS being TopSpace
for SS being non empty TopSpace
for f being Function of TS,SS st TS is compact & SS is_T2 & rng f = [#] SS & f is continuous holds
for PS being Subset of TS st PS is closed holds
f .: PS is closed
proof end;

theorem Th26: :: COMPTS_1:26
for TS being TopSpace
for SS being non empty TopSpace
for f being Function of TS,SS st TS is compact & SS is_T2 & dom f = [#] TS & rng f = [#] SS & f is one-to-one & f is continuous holds
f is being_homeomorphism
proof end;

definition
let D be set ;
func 1TopSp c1 -> TopStruct equals :: COMPTS_1:def 8
TopStruct(# D,([#] (bool D)) #);
coherence
TopStruct(# D,([#] (bool D)) #) is TopStruct
;
end;

:: deftheorem Def8 defines 1TopSp COMPTS_1:def 8 :
for D being set holds 1TopSp D = TopStruct(# D,([#] (bool D)) #);

registration
let D be set ;
cluster 1TopSp a1 -> strict TopSpace-like ;
coherence
( 1TopSp D is strict & 1TopSp D is TopSpace-like )
proof end;
end;

registration
let D be non empty set ;
cluster 1TopSp a1 -> non empty strict TopSpace-like ;
coherence
not 1TopSp D is empty
by STRUCT_0:def 1;
end;

registration
let x be set ;
cluster 1TopSp {a1} -> non empty strict TopSpace-like being_T2 ;
coherence
1TopSp {x} is being_T2
proof end;
end;

registration
cluster non empty being_T2 TopStruct ;
existence
ex b1 being TopSpace st
( b1 is being_T2 & not b1 is empty )
proof end;
end;

registration
let T be non empty being_T2 TopSpace;
cluster compact -> closed Element of bool the carrier of a1;
coherence
for b1 being Subset of T st b1 is compact holds
b1 is closed
by ;
end;

registration
let T be 1-sorted ;
cluster finite Element of bool the carrier of a1;
existence
ex b1 being Subset of T st b1 is finite
proof end;
end;

theorem Th27: :: COMPTS_1:27
for T being TopSpace st the carrier of T is finite holds
T is compact
proof end;

registration
let T be TopSpace;
cluster finite -> compact Element of bool the carrier of a1;
coherence
for b1 being Subset of T st b1 is finite holds
b1 is compact
proof end;
end;