TOPGEN_2: On the characteristic and weight of a topological space

:: On the characteristic and weight of a topological space
:: by Grzegorz Bancerek
::
:: Received December 10, 2004
:: Copyright (c) 2004-2021 Association of Mizar Users

::<DESC name="baza w punkcie a baza" monograph="topology"/>

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;

::<DESC name="baza w punkcie a baza" monograph="topology"/>

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

proof end;

uniqueness

proof end;

end;

:: deftheorem Def1 defines Chi TOPGEN_2:def 1 :

::<DESC name="charakter punktu" monograph="topology"/>

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 :: thesis:

let T be non empty TopStruct ; :: thesis:
set X = { (Chi (x,T)) where x is Point of T : verum } ;

now :: thesis:

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

end;

hence union { (Chi (x,T)) where x is Point of T : verum } is cardinal number by Th3; :: thesis:
let m be cardinal number ; :: thesis:
assume A1: m = union { (Chi (x,T)) where x is Point of T : verum } ; :: thesis:

hereby :: thesis:

let x be Point of T; :: thesis:
Chi (x,T) in { (Chi (x,T)) where x is Point of T : verum } ;
hence Chi (x,T) c= m by A1, ZFMISC_1:74; :: thesis:

end;

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

now :: thesis:

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

end;

hence m c= M by A1, ZFMISC_1:76; :: thesis:

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

proof end;

uniqueness

proof end;

end;

:: deftheorem Def2 defines Chi TOPGEN_2:def 2 :

::<DESC name="charakter przestrzeni" monograph="topology"/>

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;

::<DESC name="pierwszy aksjomat przeliczalno/sci" monograph="topology"/>

theorem :: TOPGEN_2:6

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

proof end;

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

proof end;

end;

:: deftheorem Def3 defines Neighborhood_System TOPGEN_2:def 3 :

::<DESC name="uk/lad otocze/n" monograph="topology"/>

definition

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

proof end;

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

end;

::<DESC name="uk/lad otocze/n: (BP1)" monograph="topology"/>
:: for T being non empty TopSpace
:: for N being Neighborhood_System of T
:: for x being Element of T holds N.x is non empty &
:: for U being set st U in N.x holds x in U
:: by YELLOW_8:21; :: and clusters YELLOW12
::<DESC name="uk/lad otocze/n: (BP2)" monograph="topology"/>

theorem :: TOPGEN_2:7

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;

::<DESC name="uk/lad otocze/n: (BP3)" monograph="topology"/>

theorem :: TOPGEN_2:8

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 :: thesis:

let T be non empty TopSpace; :: thesis:
let A be Subset of T; :: thesis:
let x be Element of T; :: thesis:
assume A1: x in Cl A ; :: thesis:
let B be Basis of x; :: thesis:
let U be set ; :: thesis:
assume A2: U in B ; :: thesis:
then x in U by YELLOW_8:12;
then U meets Cl A by A1, XBOOLE_0:3;
hence U meets A by A2, TSEP_1:36, YELLOW_8:12; :: thesis:

end;

Lm3: now :: thesis:

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

now :: thesis:

let U be Subset of T; :: thesis:
assume that
A2: U is open and
A3: x in U ; :: thesis:
ex V being Subset of T st
( V in B & V c= U ) by A2, A3, YELLOW_8:def 1;
hence A meets U by A1, XBOOLE_1:63; :: thesis:

end;

hence x in Cl A by PRE_TOPC:def 7; :: thesis:

end;

::<DESC name="1.1.1. Twierdzenie: Cl a baza w punkcie" monograph="topology"/>

theorem :: TOPGEN_2:9

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;

::<DESC name="1.1.1. Twierdzenie: Cl a baza w punkcie" monograph="topology"/>

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 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 for Element of K32(K32( the carrier of T));

existence

proof end;

end;

::<DESC name="1.1.14. Twierdzenie: istnienie podrodziny mocy niewi/ekszej od wagi przestrzeni i o unii r/ownej" monograph="topology"/>

theorem Th11: :: TOPGEN_2:11

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 :

notation

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

end;

registration

cluster finite -> finite-weight for TopStruct ;

coherence

proof end;

cluster infinite-weight -> infinite for TopStruct ;

coherence ;

end;

theorem Th12: :: TOPGEN_2:12

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

proof end;

theorem Th13: :: TOPGEN_2:13

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 Th14: :: TOPGEN_2:14

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

proof end;

registration

cluster TopSpace-like infinite-weight for TopStruct ;

existence

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 Th15: :: TOPGEN_2:15

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 Th16: :: TOPGEN_2:16

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 Th17: :: TOPGEN_2:17

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;

::<DESC name="1.1.15. Twierdzenie: o istnieniu bazy bed/acej podrodzin/a bazy i maj/acej moc r/own/a wadze przestrzeni" monograph="topology"/>

theorem :: TOPGEN_2:18

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;

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 ;
existence

proof end;

end;

:: deftheorem Def5 defines DiscrWithInfin TOPGEN_2:def 5 :

registration

let X, x0 be set ;

cluster DiscrWithInfin (X,x0) -> strict TopSpace-like ;

coherence

proof end;

end;

registration

let X be non empty set ;
let x0 be set ;

cluster DiscrWithInfin (X,x0) -> non empty strict ;

coherence by Def5;

end;

theorem Th19: :: TOPGEN_2:19

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 Th20: :: TOPGEN_2:20

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_2:21

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

proof end;

theorem Th22: :: TOPGEN_2:22

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_2:23

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

proof end;

theorem Th24: :: TOPGEN_2:24

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

proof end;

theorem Th25: :: TOPGEN_2:25

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 Th26: :: TOPGEN_2:26

for X, x0 being set st x0 in X holds
for A being Subset of (DiscrWithInfin (X,x0)) st A is infinite holds
Cl A = A \/ {x0}

proof end;

theorem :: TOPGEN_2:27

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_2:28

for X, x0 being set st x0 in X holds
for A being Subset of (DiscrWithInfin (X,x0)) st A ` is infinite holds
Int A = A \ {x0}

proof end;

theorem Th29: :: TOPGEN_2:29

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_2:30

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

proof end;

theorem Th31: :: TOPGEN_2:31

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

proof end;

theorem Th32: :: TOPGEN_2:32

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 Th33: :: TOPGEN_2:33

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_2:34

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

proof end;

theorem :: TOPGEN_2:35

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;

::<DSCR name="Exercise 1.1.B" monograph="topology"/>

theorem :: TOPGEN_2:36

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;

::<DSCR name="Exercise 1.1.B" monograph="topology"/>

theorem :: TOPGEN_2:37

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;

::<DSCR name="Exercise 1.1.D" monograph="topology"/>

theorem :: TOPGEN_2:38

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;

::<DSCR name="Exercise 1.1.D" monograph="topology"/>

theorem :: TOPGEN_2:39

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;