:: COMPTS_1 semantic presentation

definition

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

end;

:: deftheorem Def1 defines is_a_cover_of COMPTS_1:def 1 :

definition

let F be set ;
attr F 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 :

definition

let T be TopStruct ;
attr T 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 T 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 T 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 T 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 :

:: deftheorem Def4 defines being_T2 COMPTS_1:def 4 :

:: deftheorem defines being_T3 COMPTS_1:def 5 :

:: deftheorem defines being_T4 COMPTS_1:def 6 :

notation

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

end;

definition

let T be TopStruct ;
let P be Subset of T;
attr P 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 :

theorem :: COMPTS_1:1

canceled;

theorem :: COMPTS_1:2

canceled;

theorem :: COMPTS_1:3

canceled;

theorem :: COMPTS_1:4

canceled;

theorem :: COMPTS_1:5

canceled;

theorem :: COMPTS_1:6

canceled;

theorem :: COMPTS_1:7

canceled;

theorem :: 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 :: 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 :: 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 :: 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 :: COMPTS_1:21

for TS being TopSpace st TS is_T2 & TS is compact holds
TS is_T3

proof end;

theorem :: COMPTS_1:22

for TS being TopSpace st TS is_T2 & TS is compact holds
TS is_T4

proof end;

theorem :: 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 :: 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;