:: Baire Spaces, Sober Spaces
:: by Andrzej Trybulec
::
:: Received January 8, 1997
:: Copyright (c) 1997-2011 Association of Mizar Users

begin

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
( X is trivial iff for x, y being Element of X holds x = y )

proof end;

end;

:: deftheorem Def1 defines trivial YELLOW_8:def 1 :
for X being non empty set holds
( X is trivial iff for x, y being Element of X holds x = y );

begin

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;

begin

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;

begin

definition

let T be non empty TopStruct ;
let x be Point of T;
let F be Subset-Family of T;
attr F is x -quasi_basis means :Def2: :: YELLOW_8:def 2
( x in Intersect F & ( for S being Subset of T st S is open & x in S holds
ex V being Subset of T st
( V in F & V c= S ) ) );

end;

:: deftheorem Def2 defines -quasi_basis YELLOW_8:def 2 :
for T being non empty TopStruct
for x being Point of T
for F being Subset-Family of T holds
( F is x -quasi_basis iff ( x in Intersect F & ( for S being Subset of T st S is open & x in S holds
ex V being Subset of T st
( V in F & V c= S ) ) ) );

registration

let T be non empty TopStruct ;
let x be Point of T;
cluster open x -quasi_basis Element of bool (bool the carrier of T);
existence
ex b₁ being Subset-Family of T st
( b₁ is open & b₁ is x -quasi_basis )

proof end;

end;

definition

let T be non empty TopStruct ;
let x be Point of T;
mode Basis of x is open x -quasi_basis Subset-Family of T;

end;

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 :
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 everywhere_dense ) holds
ex I being Subset of T st
( I = Intersect F & I is dense ) );

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;

begin

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 :
for T being TopStruct
for S being Subset of T holds
( S is irreducible iff ( 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 ) ) );

registration

let T be TopStruct ;
cluster irreducible -> non empty Element of bool the carrier of T;
coherence
for b₁ being Subset of T st b₁ is irreducible holds
not b₁ is empty 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 :
for T being non empty TopSpace
for S being Subset of T
for p being Point of T holds
( p is_dense_point_of S iff ( p in S & S c= Cl {p} ) );

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 irreducible Element of bool the carrier of T;
existence
ex b₁ being Subset of T st b₁ is irreducible

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 :
for T being non empty TopSpace holds
( T is sober iff 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 ) ) );

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
for b₁ being Subset of T st b₁ is irreducible holds
b₁ is trivial 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 TopSpace-like Hausdorff -> non empty sober TopStruct ;
coherence
for b₁ being non empty TopSpace st b₁ is T_2 holds
b₁ is sober by Th31;

end;

registration

cluster non empty TopSpace-like sober TopStruct ;
existence
ex b₁ being non empty TopSpace st b₁ is sober

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 TopSpace-like sober -> non empty T_0 TopStruct ;
coherence
for b₁ being non empty TopSpace st b₁ is sober holds
b₁ is T_0 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
ex b₁ being strict TopStruct st
( the carrier of b₁ = X & COMPLEMENT the topology of b₁ = {X} \/ (Fin X) )

proof end;

uniqueness
for b₁, b₂ being strict TopStruct st the carrier of b₁ = X & COMPLEMENT the topology of b₁ = {X} \/ (Fin X) & the carrier of b₂ = X & COMPLEMENT the topology of b₂ = {X} \/ (Fin X) holds
b₁ = b₂ by SETFAM_1:53;

end;

:: deftheorem Def7 defines CofinTop YELLOW_8:def 7 :
for X being set
for b₂ being strict TopStruct holds
( b₂ = CofinTop X iff ( the carrier of b₂ = X & COMPLEMENT the topology of b₂ = {X} \/ (Fin X) ) );

registration

let X be non empty set ;
cluster CofinTop X -> non empty strict ;
coherence
not CofinTop X is empty by Def7;

end;

registration

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

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 T_1 holds
for p being Point of T holds Cl {p} = {p}

proof end;

registration

let X be non empty set ;
cluster CofinTop X -> strict T_1 ;
coherence
CofinTop X is T_1

proof end;

end;

registration

let X be infinite set ;
cluster CofinTop X -> strict non sober ;
coherence
not CofinTop X is sober

proof end;

end;

registration

cluster non empty TopSpace-like T_1 non sober TopStruct ;
existence
ex b₁ being non empty TopSpace st
( b₁ is T_1 & not b₁ is sober )

proof end;

end;

begin

theorem Th36: :: YELLOW_8:36

for T being non empty TopSpace holds
( T is regular 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 regular 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;