:: YELLOW_8 semantic presentation

theorem Th1: :: YELLOW_8:1

for X, A, B being set st A in Fin X & B c= A holds
B in Fin X

proof end;

theorem Th2: :: YELLOW_8:2

for X being set
for F being Subset-Family of X st F c= Fin X holds
meet F in Fin X

proof end;

definition

let X be non empty set ;
redefine attr X is trivial means :Def1: :: YELLOW_8:def 1
for x, y being Element of X holds x = y;
compatibility

proof end;

end;

:: deftheorem Def1 defines trivial YELLOW_8:def 1 :

theorem :: YELLOW_8:3

canceled;

theorem Th4: :: YELLOW_8:4

for X being set
for F being Subset-Family of X holds F, COMPLEMENT F are_equipotent

proof end;

theorem Th5: :: YELLOW_8:5

for X, Y being set st X,Y are_equipotent & X is countable holds
Y is countable

proof end;

theorem :: YELLOW_8:6

canceled;

theorem :: YELLOW_8:7

canceled;

theorem :: YELLOW_8:8

canceled;

theorem :: YELLOW_8:9

canceled;

theorem :: YELLOW_8:10

canceled;

theorem :: YELLOW_8:11

canceled;

theorem :: YELLOW_8:12

canceled;

theorem :: YELLOW_8:13

canceled;

theorem :: YELLOW_8:14

for X being 1-sorted
for F being Subset-Family of X
for P being Subset of X holds
( P ` in COMPLEMENT F iff P in F )

proof end;

theorem Th15: :: YELLOW_8:15

for X being 1-sorted
for F being Subset-Family of X holds Intersect (COMPLEMENT F) = (union F) `

proof end;

theorem Th16: :: YELLOW_8:16

for X being 1-sorted
for F being Subset-Family of X holds union (COMPLEMENT F) = (Intersect F) `

proof end;

theorem :: YELLOW_8:17

for T being non empty TopSpace
for A, B being Subset of T st B c= A & A is closed & ( for C being Subset of T st B c= C & C is closed holds
A c= C ) holds
A = Cl B

proof end;

theorem Th18: :: YELLOW_8:18

for T being TopStruct
for B being Basis of T
for V being Subset of T st V is open holds
V = union { G where G is Subset of T : ( G in B & G c= V ) }

proof end;

theorem Th19: :: YELLOW_8:19

for T being TopStruct
for B being Basis of T
for S being Subset of T st S in B holds
S is open

proof end;

theorem :: YELLOW_8:20

for T being non empty TopSpace
for B being Basis of T
for V being Subset of T holds Int V = union { G where G is Subset of T : ( G in B & G c= V ) }

proof end;

definition

let T be non empty TopStruct ;
let x be Point of T;
mode Basis of x -> Subset-Family of T means :Def2: :: YELLOW_8:def 2
( it c= the topology of T & x in Intersect it & ( for S being Subset of T st S is open & x in S holds
ex V being Subset of T st
( V in it & V c= S ) ) );
existence

proof end;

end;

:: deftheorem Def2 defines Basis YELLOW_8:def 2 :

theorem Th21: :: YELLOW_8:21

for T being non empty TopStruct
for x being Point of T
for B being Basis of x
for V being Subset of T st V in B holds
( V is open & x in V )

proof end;

theorem :: YELLOW_8:22

for T being non empty TopStruct
for x being Point of T
for B being Basis of x
for V being Subset of T st x in V & V is open holds
ex W being Subset of T st
( W in B & W c= V ) by Def2;

theorem :: YELLOW_8:23

for T being non empty TopStruct
for P being Subset-Family of T st P c= the topology of T & ( for x being Point of T ex B being Basis of x st B c= P ) holds
P is Basis of T

proof end;

definition

let T be non empty TopSpace;
attr T is Baire means :Def3: :: YELLOW_8:def 3
for F being Subset-Family of T st F is countable & ( for S being Subset of T st S in F holds
S is everywhere_dense ) holds
ex I being Subset of T st
( I = Intersect F & I is dense );

end;

:: deftheorem Def3 defines Baire YELLOW_8:def 3 :

theorem :: YELLOW_8:24

for T being non empty TopSpace holds
( T is Baire iff for F being Subset-Family of T st F is countable & ( for S being Subset of T st S in F holds
S is nowhere_dense ) holds
union F is boundary )

proof end;

definition

let T be TopStruct ;
let S be Subset of T;
attr S is irreducible means :Def4: :: YELLOW_8:def 4
( not S is empty & S is closed & ( for S1, S2 being Subset of T st S1 is closed & S2 is closed & S = S1 \/ S2 & not S1 = S holds
S2 = S ) );

end;

:: deftheorem Def4 defines irreducible YELLOW_8:def 4 :

registration

let T be TopStruct ;
cluster irreducible -> non empty Element of bool the carrier of T;
coherence by Def4;

end;

definition

let T be non empty TopSpace;
let S be Subset of T;
let p be Point of T;
pred p is_dense_point_of S means :Def5: :: YELLOW_8:def 5
( p in S & S c= Cl {p} );

end;

:: deftheorem Def5 defines is_dense_point_of YELLOW_8:def 5 :

theorem :: YELLOW_8:25

for T being non empty TopSpace
for S being Subset of T st S is closed holds
for p being Point of T st p is_dense_point_of S holds
S = Cl {p}

proof end;

theorem Th26: :: YELLOW_8:26

for T being non empty TopSpace
for p being Point of T holds Cl {p} is irreducible

proof end;

registration

let T be non empty TopSpace;
cluster non empty irreducible Element of bool the carrier of T;
existence

proof end;

end;

definition

let T be non empty TopSpace;
attr T is sober means :Def6: :: YELLOW_8:def 6
for S being irreducible Subset of T ex p being Point of T st
( p is_dense_point_of S & ( for q being Point of T st q is_dense_point_of S holds
p = q ) );

end;

:: deftheorem Def6 defines sober YELLOW_8:def 6 :

theorem Th27: :: YELLOW_8:27

for T being non empty TopSpace
for p being Point of T holds p is_dense_point_of Cl {p}

proof end;

theorem Th28: :: YELLOW_8:28

for T being non empty TopSpace
for p being Point of T holds p is_dense_point_of {p}

proof end;

theorem Th29: :: YELLOW_8:29

for T being non empty TopSpace
for G, F being Subset of T st G is open & F is closed holds
F \ G is closed

proof end;

theorem Th30: :: YELLOW_8:30

for T being non empty Hausdorff TopSpace
for S being irreducible Subset of T holds S is trivial

proof end;

registration

let T be non empty Hausdorff TopSpace;
cluster irreducible -> trivial Element of bool the carrier of T;
coherence by Th30;

end;

theorem Th31: :: YELLOW_8:31

for T being non empty Hausdorff TopSpace holds T is sober

proof end;

registration

cluster non empty Hausdorff -> non empty sober TopStruct ;
coherence by Th31;

end;

registration

cluster non empty sober TopStruct ;
existence

proof end;

end;

theorem Th32: :: YELLOW_8:32

for T being non empty TopSpace holds
( T is T_0 iff for p, q being Point of T st Cl {p} = Cl {q} holds
p = q )

proof end;

theorem Th33: :: YELLOW_8:33

for T being non empty sober TopSpace holds T is T_0

proof end;

registration

cluster non empty sober -> non empty T_0 TopStruct ;
coherence by Th33;

end;

definition

let X be set ;
func CofinTop X -> strict TopStruct means :Def7: :: YELLOW_8:def 7
( the carrier of it = X & COMPLEMENT the topology of it = {X} \/ (Fin X) );
existence

proof end;

uniqueness by SETFAM_1:53;

end;

:: deftheorem Def7 defines CofinTop YELLOW_8:def 7 :

registration

let X be non empty set ;
cluster CofinTop X -> non empty strict ;
coherence

proof end;

end;

registration

let X be set ;
cluster CofinTop X -> strict TopSpace-like ;
coherence

proof end;

end;

theorem Th34: :: YELLOW_8:34

for X being non empty set
for P being Subset of (CofinTop X) holds
( P is closed iff ( P = X or P is finite ) )

proof end;

theorem Th35: :: YELLOW_8:35

for T being non empty TopSpace st T is_T1 holds
for p being Point of T holds Cl {p} = {p}

proof end;

registration

let X be non empty set ;
cluster CofinTop X -> non empty strict TopSpace-like being_T1 ;
coherence

proof end;

end;

registration

let X be infinite set ;
cluster CofinTop X -> non empty strict TopSpace-like being_T1 non sober ;
coherence

proof end;

end;

registration

cluster non empty being_T1 non sober TopStruct ;
existence

proof end;

end;

theorem Th36: :: YELLOW_8:36

for T being non empty TopSpace holds
( T is_T3 iff for p being Point of T
for P being Subset of T st p in Int P holds
ex Q being Subset of T st
( Q is closed & Q c= P & p in Int Q ) )

proof end;

theorem :: YELLOW_8:37

for T being non empty TopSpace st T is_T3 holds
( T is locally-compact iff for x being Point of T ex Y being Subset of T st
( x in Int Y & Y is compact ) )

proof end;