:: On the characteristic and weight of a topological space
:: by Grzegorz Bancerek
::
:: Received December 10, 2004
:: Copyright (c) 2004-2011 Association of Mizar Users

begin

theorem Th1: :: TOPGEN_2:1

for T being non empty TopSpace
for B being Basis of T
for x being Element of T holds { U where U is Subset of T : ( x in U & U in B ) } is Basis of x

proof end;

theorem Th2: :: TOPGEN_2:2

for T being non empty TopSpace
for B being ManySortedSet of T st ( for x being Element of T holds B . x is Basis of x ) holds
Union B is Basis of T

proof end;

definition

let T be non empty TopStruct ;
let x be Element of T;
func Chi (x,T) -> cardinal number means :Def1: :: TOPGEN_2:def 1
( ex B being Basis of x st it = card B & ( for B being Basis of x holds it c= card B ) );
existence
ex b₁ being cardinal number st
( ex B being Basis of x st b₁ = card B & ( for B being Basis of x holds b₁ c= card B ) )

proof end;

uniqueness
for b₁, b₂ being cardinal number st ex B being Basis of x st b₁ = card B & ( for B being Basis of x holds b₁ c= card B ) & ex B being Basis of x st b₂ = card B & ( for B being Basis of x holds b₂ c= card B ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def1 defines Chi TOPGEN_2:def 1 :
for T being non empty TopStruct
for x being Element of T
for b₃ being cardinal number holds
( b₃ = Chi (x,T) iff ( ex B being Basis of x st b₃ = card B & ( for B being Basis of x holds b₃ c= card B ) ) );

theorem Th3: :: TOPGEN_2:3

for X being set st ( for a being set st a in X holds
a is cardinal number ) holds
union X is cardinal number

proof end;

Lm1: now

let T be non empty TopStruct ; :: thesis: ( union { (Chi (x,T)) where x is Point of T : verum } is cardinal number & ( for m being cardinal number st m = union { (Chi (x,T)) where x is Point of T : verum } holds
( ( for x being Point of T holds Chi (x,T) c= m ) & ( for M being cardinal number st ( for x being Point of T holds Chi (x,T) c= M ) holds
m c= M ) ) ) )
set X = { (Chi (x,T)) where x is Point of T : verum } ;

now

let a be set ; :: thesis: ( a in { (Chi (x,T)) where x is Point of T : verum } implies a is cardinal number )
assume a in { (Chi (x,T)) where x is Point of T : verum } ; :: thesis: a is cardinal number
then ex x being Point of T st a = Chi (x,T) ;
hence a is cardinal number ; :: thesis: verum

end;

hence union { (Chi (x,T)) where x is Point of T : verum } is cardinal number by Th3; :: thesis: for m being cardinal number st m = union { (Chi (x,T)) where x is Point of T : verum } holds
( ( for x being Point of T holds Chi (x,T) c= m ) & ( for M being cardinal number st ( for x being Point of T holds Chi (x,T) c= M ) holds
m c= M ) )
let m be cardinal number ; :: thesis: ( m = union { (Chi (x,T)) where x is Point of T : verum } implies ( ( for x being Point of T holds Chi (x,T) c= m ) & ( for M being cardinal number st ( for x being Point of T holds Chi (x,T) c= M ) holds
m c= M ) ) )
assume A1: m = union { (Chi (x,T)) where x is Point of T : verum } ; :: thesis: ( ( for x being Point of T holds Chi (x,T) c= m ) & ( for M being cardinal number st ( for x being Point of T holds Chi (x,T) c= M ) holds
m c= M ) )

hereby :: thesis: for M being cardinal number st ( for x being Point of T holds Chi (x,T) c= M ) holds
m c= M

let x be Point of T; :: thesis: Chi (x,T) c= m
Chi (x,T) in { (Chi (x,T)) where x is Point of T : verum } ;
hence Chi (x,T) c= m by A1, ZFMISC_1:92; :: thesis: verum

end;

let M be cardinal number ; :: thesis: ( ( for x being Point of T holds Chi (x,T) c= M ) implies m c= M )
assume A2: for x being Point of T holds Chi (x,T) c= M ; :: thesis: m c= M

now

let a be set ; :: thesis: ( a in { (Chi (x,T)) where x is Point of T : verum } implies a c= M )
assume a in { (Chi (x,T)) where x is Point of T : verum } ; :: thesis: a c= M
then ex x being Point of T st a = Chi (x,T) ;
hence a c= M by A2; :: thesis: verum

end;

hence m c= M by A1, ZFMISC_1:94; :: thesis: verum

end;

definition

let T be non empty TopStruct ;
func Chi T -> cardinal number means :Def2: :: TOPGEN_2:def 2
( ( for x being Point of T holds Chi (x,T) c= it ) & ( for M being cardinal number st ( for x being Point of T holds Chi (x,T) c= M ) holds
it c= M ) );
existence
ex b₁ being cardinal number st
( ( for x being Point of T holds Chi (x,T) c= b₁ ) & ( for M being cardinal number st ( for x being Point of T holds Chi (x,T) c= M ) holds
b₁ c= M ) )

proof end;

uniqueness
for b₁, b₂ being cardinal number st ( for x being Point of T holds Chi (x,T) c= b₁ ) & ( for M being cardinal number st ( for x being Point of T holds Chi (x,T) c= M ) holds
b₁ c= M ) & ( for x being Point of T holds Chi (x,T) c= b₂ ) & ( for M being cardinal number st ( for x being Point of T holds Chi (x,T) c= M ) holds
b₂ c= M ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def2 defines Chi TOPGEN_2:def 2 :
for T being non empty TopStruct
for b₂ being cardinal number holds
( b₂ = Chi T iff ( ( for x being Point of T holds Chi (x,T) c= b₂ ) & ( for M being cardinal number st ( for x being Point of T holds Chi (x,T) c= M ) holds
b₂ c= M ) ) );

theorem Th4: :: TOPGEN_2:4

for T being non empty TopStruct holds Chi T = union { (Chi (x,T)) where x is Point of T : verum }

proof end;

theorem Th5: :: TOPGEN_2:5

for T being non empty TopStruct
for x being Point of T holds Chi (x,T) c= Chi T

proof end;

theorem :: TOPGEN_2:6

for T being non empty TopSpace holds
( T is first-countable iff Chi T c= omega )

proof end;

begin

definition

let T be non empty TopSpace;
mode Neighborhood_System of T -> ManySortedSet of T means :Def3: :: TOPGEN_2:def 3
for x being Element of T holds it . x is Basis of x;
existence
ex b₁ being ManySortedSet of T st
for x being Element of T holds b₁ . x is Basis of x

proof end;

end;

:: deftheorem Def3 defines Neighborhood_System TOPGEN_2:def 3 :
for T being non empty TopSpace
for b₂ being ManySortedSet of T holds
( b₂ is Neighborhood_System of T iff for x being Element of T holds b₂ . x is Basis of x );

definition

let T be non empty TopSpace;
let N be Neighborhood_System of T;
:: original: Union
redefine func Union N -> Basis of T;
coherence
Union N is Basis of T

proof end;

let x be Point of T;
:: original: .
redefine func N . x -> Basis of x;
coherence
N . x is Basis of x by Def3;

end;

theorem :: TOPGEN_2:7

canceled;

theorem :: TOPGEN_2:8

for T being non empty TopSpace
for x, y being Point of T
for B1 being Basis of x
for B2 being Basis of y
for U being set st x in U & U in B2 holds
ex V being open Subset of T st
( V in B1 & V c= U )

proof end;

theorem :: TOPGEN_2:9

for T being non empty TopSpace
for x being Point of T
for B being Basis of x
for U1, U2 being set st U1 in B & U2 in B holds
ex V being open Subset of T st
( V in B & V c= U1 /\ U2 )

proof end;

Lm2: now

let T be non empty TopSpace; :: thesis: for A being Subset of T
for x being Element of T st x in Cl A holds
for B being Basis of x
for U being set st U in B holds
U meets A
let A be Subset of T; :: thesis: for x being Element of T st x in Cl A holds
for B being Basis of x
for U being set st U in B holds
U meets A
let x be Element of T; :: thesis: ( x in Cl A implies for B being Basis of x
for U being set st U in B holds
U meets A )
assume A1: x in Cl A ; :: thesis: for B being Basis of x
for U being set st U in B holds
U meets A
let B be Basis of x; :: thesis: for U being set st U in B holds
U meets A
let U be set ; :: thesis: ( U in B implies U meets A )
assume A2: U in B ; :: thesis: U meets A
then x in U by YELLOW_8:21;
then U meets Cl A by A1, XBOOLE_0:3;
hence U meets A by A2, TSEP_1:40, YELLOW_8:21; :: thesis: verum

end;

Lm3: now

let T be non empty TopSpace; :: thesis: for A being Subset of T
for x being Element of T st ex B being Basis of x st
for U being set st U in B holds
U meets A holds
x in Cl A
let A be Subset of T; :: thesis: for x being Element of T st ex B being Basis of x st
for U being set st U in B holds
U meets A holds
x in Cl A
let x be Element of T; :: thesis: ( ex B being Basis of x st
for U being set st U in B holds
U meets A implies x in Cl A )
given B being Basis of x such that A1: for U being set st U in B holds
U meets A ; :: thesis: x in Cl A

now

let U be Subset of T; :: thesis: ( U is open & x in U implies A meets U )
assume that
A2: U is open and
A3: x in U ; :: thesis: A meets U
ex V being Subset of T st
( V in B & V c= U ) by A2, A3, YELLOW_8:def 2;
hence A meets U by A1, XBOOLE_1:63; :: thesis: verum

end;

hence x in Cl A by PRE_TOPC:def 13; :: thesis: verum

end;

theorem :: TOPGEN_2:10

for T being non empty TopSpace
for A being Subset of T
for x being Element of T holds
( x in Cl A iff for B being Basis of x
for U being set st U in B holds
U meets A )

proof end;

theorem :: TOPGEN_2:11

for T being non empty TopSpace
for A being Subset of T
for x being Element of T holds
( x in Cl A iff ex B being Basis of x st
for U being set st U in B holds
U meets A )

proof end;

registration

let T be TopSpace;
cluster non empty open Element of K10(K10( the carrier of T));
existence
ex b₁ being Subset-Family of T st
( b₁ is open & not b₁ is empty )

proof end;

end;

begin

theorem Th12: :: TOPGEN_2:12

for T being non empty TopSpace
for S being open Subset-Family of T ex G being open Subset-Family of T st
( G c= S & union G = union S & card G c= weight T )

proof end;

definition

let T be TopStruct ;
attr T is finite-weight means :Def4: :: TOPGEN_2:def 4
weight T is finite ;

end;

:: deftheorem Def4 defines finite-weight TOPGEN_2:def 4 :
for T being TopStruct holds
( T is finite-weight iff weight T is finite );

notation

let T be TopStruct ;
antonym infinite-weight T for finite-weight ;

end;

registration

cluster finite -> finite-weight TopStruct ;
coherence
for b₁ being TopStruct st b₁ is finite holds
b₁ is finite-weight

proof end;

cluster infinite-weight -> infinite TopStruct ;
coherence
for b₁ being TopStruct st not b₁ is finite-weight holds
not b₁ is finite ;

end;

theorem Th13: :: TOPGEN_2:13

for X being set holds card (SmallestPartition X) = card X

proof end;

theorem Th14: :: TOPGEN_2:14

for T being non empty discrete TopStruct holds
( SmallestPartition the carrier of T is Basis of T & ( for B being Basis of T holds SmallestPartition the carrier of T c= B ) )

proof end;

theorem Th15: :: TOPGEN_2:15

for T being non empty discrete TopStruct holds weight T = card the carrier of T

proof end;

registration

cluster TopSpace-like infinite-weight TopStruct ;
existence
not for b₁ being TopSpace holds b₁ is finite-weight

proof end;

end;

Lm4: for T being infinite-weight TopSpace
for B being Basis of T ex B1 being Basis of T st
( B1 c= B & card B1 = weight T )

proof end;

theorem :: TOPGEN_2:16

canceled;

theorem Th17: :: TOPGEN_2:17

for T being non empty finite-weight TopSpace ex B0 being Basis of T ex f being Function of the carrier of T, the topology of T st
( B0 = rng f & ( for x being Point of T holds
( x in f . x & ( for U being open Subset of T st x in U holds
f . x c= U ) ) ) )

proof end;

theorem Th18: :: TOPGEN_2:18

for T being TopSpace
for B0, B being Basis of T
for f being Function of the carrier of T, the topology of T st B0 = rng f & ( for x being Point of T holds
( x in f . x & ( for U being open Subset of T st x in U holds
f . x c= U ) ) ) holds
B0 c= B

proof end;

theorem Th19: :: TOPGEN_2:19

for T being TopSpace
for B0 being Basis of T
for f being Function of the carrier of T, the topology of T st B0 = rng f & ( for x being Point of T holds
( x in f . x & ( for U being open Subset of T st x in U holds
f . x c= U ) ) ) holds
weight T = card B0

proof end;

Lm5: for T being non empty finite-weight TopSpace
for B being Basis of T ex B1 being Basis of T st
( B1 c= B & card B1 = weight T )

proof end;

theorem :: TOPGEN_2:20

for T being non empty TopSpace
for B being Basis of T ex B1 being Basis of T st
( B1 c= B & card B1 = weight T )

proof end;

begin

definition

let X, x0 be set ;
func DiscrWithInfin (X,x0) -> strict TopStruct means :Def5: :: TOPGEN_2:def 5
( the carrier of it = X & the topology of it = { U where U is Subset of X : not x0 in U } \/ { (F `) where F is Subset of X : F is finite } );
uniqueness
for b<sub>1</sub>, b<sub>2</sub> being strict TopStruct st the carrier of b<sub>1</sub> = X & the topology of b<sub>1</sub> = { U where U is Subset of X : not x0 in U } \/ { (F `) where F is Subset of X : F is finite } & the carrier of b₂ = X & the topology of b₂ = { U where U is Subset of X : not x0 in U } \/ { (F `) where F is Subset of X : F is finite } holds
b<sub>1</sub> = b<sub>2</sub> ;
existence
ex b<sub>1</sub> being strict TopStruct st
( the carrier of b<sub>1</sub> = X & the topology of b<sub>1</sub> = { U where U is Subset of X : not x0 in U } \/ { (F `) where F is Subset of X : F is finite } )

proof end;

end;

:: deftheorem Def5 defines DiscrWithInfin TOPGEN_2:def 5 :
for X, x0 being set
for b₃ being strict TopStruct holds
( b₃ = DiscrWithInfin (X,x0) iff ( the carrier of b₃ = X & the topology of b₃ = { U where U is Subset of X : not x0 in U } \/ { (F `) where F is Subset of X : F is finite } ) );

registration

let X, x0 be set ;
cluster DiscrWithInfin (X,x0) -> strict TopSpace-like ;
coherence
DiscrWithInfin (X,x0) is TopSpace-like

proof end;

end;

registration

let X be non empty set ;
let x0 be set ;
cluster DiscrWithInfin (X,x0) -> non empty strict ;
coherence
not DiscrWithInfin (X,x0) is empty by Def5;

end;

theorem Th21: :: TOPGEN_2:21

for X, x0 being set
for A being Subset of (DiscrWithInfin (X,x0)) holds
( A is open iff ( not x0 in A or A ` is finite ) )

proof end;

theorem Th22: :: TOPGEN_2:22

for X, x0 being set
for A being Subset of (DiscrWithInfin (X,x0)) holds
( A is closed iff not ( x0 in X & not x0 in A & not A is finite ) )

proof end;

theorem :: TOPGEN_2:23

for X, x0, x being set st x in X holds
{x} is closed Subset of (DiscrWithInfin (X,x0))

proof end;

theorem Th24: :: TOPGEN_2:24

for X, x0, x being set st x in X & x <> x0 holds
{x} is open Subset of (DiscrWithInfin (X,x0))

proof end;

theorem :: TOPGEN_2:25

for X, x0 being set st not X is finite holds
for U being Subset of (DiscrWithInfin (X,x0)) st U = {x0} holds
not U is open

proof end;

theorem Th26: :: TOPGEN_2:26

for X, x0 being set
for A being Subset of (DiscrWithInfin (X,x0)) st A is finite holds
Cl A = A

proof end;

theorem Th27: :: TOPGEN_2:27

for T being non empty TopSpace
for A being Subset of T st not A is closed holds
for a being Point of T st A \/ {a} is closed holds
Cl A = A \/ {a}

proof end;

theorem Th28: :: TOPGEN_2:28

for X, x0 being set st x0 in X holds
for A being Subset of (DiscrWithInfin (X,x0)) st not A is finite holds
Cl A = A \/ {x0}

proof end;

theorem :: TOPGEN_2:29

for X, x0 being set
for A being Subset of (DiscrWithInfin (X,x0)) st A ` is finite holds
Int A = A

proof end;

theorem :: TOPGEN_2:30

for X, x0 being set st x0 in X holds
for A being Subset of (DiscrWithInfin (X,x0)) st not A ` is finite holds
Int A = A \ {x0}

proof end;

theorem Th31: :: TOPGEN_2:31

for X, x0 being set ex B0 being Basis of (DiscrWithInfin (X,x0)) st B0 = ((SmallestPartition X) \ {{x0}}) \/ { (F `) where F is Subset of X : F is finite }

proof end;

theorem :: TOPGEN_2:32

for X being infinite set holds card (Fin X) = card X

proof end;

theorem Th33: :: TOPGEN_2:33

for X being infinite set holds card { (F `) where F is Subset of X : F is finite } = card X

proof end;

theorem Th34: :: TOPGEN_2:34

for X being infinite set
for x0 being set
for B0 being Basis of (DiscrWithInfin (X,x0)) st B0 = ((SmallestPartition X) \ {{x0}}) \/ { (F `) where F is Subset of X : F is finite } holds
card B0 = card X

proof end;

theorem Th35: :: TOPGEN_2:35

for X being infinite set
for x0 being set
for B being Basis of (DiscrWithInfin (X,x0)) holds card X c= card B

proof end;

theorem :: TOPGEN_2:36

for X being infinite set
for x0 being set holds weight (DiscrWithInfin (X,x0)) = card X

proof end;

theorem :: TOPGEN_2:37

for X being non empty set
for x0 being set ex B0 being prebasis of (DiscrWithInfin (X,x0)) st B0 = ((SmallestPartition X) \ {{x0}}) \/ { ({x} `) where x is Element of X : verum }

proof end;

begin

theorem :: TOPGEN_2:38

for T being TopSpace
for F being Subset-Family of T
for I being non empty Subset-Family of F st ( for G being set st G in I holds
F \ G is finite ) holds
Cl (union F) = (union (clf F)) \/ (meet { (Cl (union G)) where G is Subset-Family of T : G in I } )

proof end;

theorem :: TOPGEN_2:39

for T being TopSpace
for F being Subset-Family of T holds Cl (union F) = (union (clf F)) \/ (meet { (Cl (union G)) where G is Subset-Family of T : ( G c= F & F \ G is finite ) } )

proof end;

theorem :: TOPGEN_2:40

for X being set
for O being Subset-Family of (bool X) st ( for o being Subset-Family of X st o in O holds
TopStruct(# X,o #) is TopSpace ) holds
ex B being Subset-Family of X st
( B = Intersect O & TopStruct(# X,B #) is TopSpace & ( for o being Subset-Family of X st o in O holds
TopStruct(# X,o #) is TopExtension of TopStruct(# X,B #) ) & ( for T being TopSpace st the carrier of T = X & ( for o being Subset-Family of X st o in O holds
TopStruct(# X,o #) is TopExtension of T ) holds
TopStruct(# X,B #) is TopExtension of T ) )

proof end;

theorem :: TOPGEN_2:41

for X being set
for O being Subset-Family of (bool X) ex B being Subset-Family of X st
( B = UniCl (FinMeetCl (union O)) & TopStruct(# X,B #) is TopSpace & ( for o being Subset-Family of X st o in O holds
TopStruct(# X,B #) is TopExtension of TopStruct(# X,o #) ) & ( for T being TopSpace st the carrier of T = X & ( for o being Subset-Family of X st o in O holds
T is TopExtension of TopStruct(# X,o #) ) holds
T is TopExtension of TopStruct(# X,B #) ) )

proof end;