:: Baire Spaces, Sober Spaces
:: by Andrzej Trybulec
::
:: Copyright (c) 1997-2021 Association of Mizar Users

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;

theorem Th3: :: YELLOW_8:3
for X being set
for F being Subset-Family of X holds F, COMPLEMENT F are_equipotent
proof end;

theorem Th4: :: YELLOW_8:4
for X, Y being set st X,Y are_equipotent & X is countable holds
Y is countable
proof end;

theorem :: YELLOW_8:5
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 Th6: :: YELLOW_8:6
for X being 1-sorted
for F being Subset-Family of X holds Intersect () = ()
proof end;

theorem Th7: :: YELLOW_8:7
for X being 1-sorted
for F being Subset-Family of X holds union () = () `
proof end;

theorem :: YELLOW_8:8
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 by ;

theorem Th9: :: YELLOW_8:9
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 Th10: :: YELLOW_8:10
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_8:11
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;
let F be Subset-Family of T;
attr F is x -quasi_basis means :Def1: :: YELLOW_8:def 1
( 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 Def1 defines -quasi_basis YELLOW_8:def 1 :
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 for 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 Th12: :: YELLOW_8:12
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_8:13
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 Def1;

theorem :: YELLOW_8:14
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 :: YELLOW_8:def 2
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 defines Baire YELLOW_8:def 2 :
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_8:15
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 :Def3: :: YELLOW_8:def 3
( 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 Def3 defines irreducible YELLOW_8:def 3 :
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 for Element of bool the carrier of T;
coherence
for b1 being Subset of T st b1 is irreducible holds
not b1 is empty
;
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 :: YELLOW_8:def 4
( p in S & S c= Cl {p} );
end;

:: deftheorem defines is_dense_point_of YELLOW_8:def 4 :
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_8:16
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} by ;

theorem Th17: :: YELLOW_8:17
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 for 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 :Def5: :: YELLOW_8:def 5
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 Def5 defines sober YELLOW_8:def 5 :
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 Th18: :: YELLOW_8:18
for T being non empty TopSpace
for p being Point of T holds p is_dense_point_of Cl {p}
proof end;

theorem Th19: :: YELLOW_8:19
for T being non empty TopSpace
for p being Point of T holds p is_dense_point_of {p} by ;

theorem Th20: :: YELLOW_8:20
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 Th21: :: YELLOW_8:21
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 for Element of bool the carrier of T;
coherence
for b1 being Subset of T st b1 is irreducible holds
b1 is trivial
by Th21;
end;

theorem Th22: :: YELLOW_8:22
for T being non empty Hausdorff TopSpace holds T is sober
proof end;

registration
coherence
for b1 being non empty TopSpace st b1 is Hausdorff holds
b1 is sober
by Th22;
end;

registration
existence
ex b1 being non empty TopSpace st b1 is sober
proof end;
end;

theorem Th23: :: YELLOW_8:23
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 Th24: :: YELLOW_8:24
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 for TopStruct ;
coherence
for b1 being non empty TopSpace st b1 is sober holds
b1 is T_0
by Th24;
end;

definition
let X be set ;
func CofinTop X -> strict TopStruct means :Def6: :: YELLOW_8:def 6
( 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_1:38;
end;

:: deftheorem Def6 defines CofinTop YELLOW_8:def 6 :
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 Def6;
end;

registration
let X be set ;
coherence
proof end;
end;

theorem Th25: :: YELLOW_8:25
for X being non empty set
for P being Subset of () holds
( P is closed iff ( P = X or P is finite ) )
proof end;

theorem Th26: :: YELLOW_8:26
for T being non empty TopSpace st T is T_1 holds
for p being Point of T holds Cl {p} = {p} by ;

registration
let X be non empty set ;
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
existence
ex b1 being non empty TopSpace st
( b1 is T_1 & not b1 is sober )
proof end;
end;

theorem Th27: :: YELLOW_8:27
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_8:28
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;