:: The Cantor Set
:: by Alexander Yu. Shibakov and Andrzej Trybulec
::
:: Received January 9, 1995
:: Copyright (c) 1995-2021 Association of Mizar Users

definition

let X be set ;
let A be Subset-Family of X;

func UniCl A -> Subset-Family of X means :Def1: :: CANTOR_1:def 1
for x being Subset of X holds
( x in it iff ex Y being Subset-Family of X st
( Y c= A & x = union Y ) );

existence
ex b₁ being Subset-Family of X st
for x being Subset of X holds
( x in b₁ iff ex Y being Subset-Family of X st
( Y c= A & x = union Y ) )

proof end;

uniqueness
for b₁, b₂ being Subset-Family of X st ( for x being Subset of X holds
( x in b₁ iff ex Y being Subset-Family of X st
( Y c= A & x = union Y ) ) ) & ( for x being Subset of X holds
( x in b₂ iff ex Y being Subset-Family of X st
( Y c= A & x = union Y ) ) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def1 defines UniCl CANTOR_1:def 1 :
for X being set
for A, b₃ being Subset-Family of X holds
( b₃ = UniCl A iff for x being Subset of X holds
( x in b₃ iff ex Y being Subset-Family of X st
( Y c= A & x = union Y ) ) );

definition

let X be TopStruct ;
let F be Subset-Family of X;

attr F is quasi_basis means :Def2: :: CANTOR_1:def 2
the topology of X c= UniCl F;

end;

:: deftheorem Def2 defines quasi_basis CANTOR_1:def 2 :
for X being TopStruct
for F being Subset-Family of X holds
( F is quasi_basis iff the topology of X c= UniCl F );

registration

let X be TopStruct ;

cluster the topology of X -> quasi_basis ;

coherence
the topology of X is quasi_basis

proof end;

end;

registration

let X be TopStruct ;

cluster open quasi_basis for Element of bool (bool the carrier of X);

existence
ex b₁ being Subset-Family of X st
( b₁ is open & b₁ is quasi_basis )

proof end;

end;

definition

let X be TopStruct ;
mode Basis of X is open quasi_basis Subset-Family of X;

end;

theorem Th1: :: CANTOR_1:1

for X being set
for A being Subset-Family of X holds A c= UniCl A

proof end;

theorem :: CANTOR_1:2

for S being TopStruct holds the topology of S is Basis of S ;

theorem :: CANTOR_1:3

for S being TopStruct holds the topology of S is open ;

definition

let X be set ;
let A be Subset-Family of X;

func FinMeetCl A -> Subset-Family of X means :Def3: :: CANTOR_1:def 3
for x being Subset of X holds
( x in it iff ex Y being Subset-Family of X st
( Y c= A & Y is finite & x = Intersect Y ) );

existence
ex b₁ being Subset-Family of X st
for x being Subset of X holds
( x in b₁ iff ex Y being Subset-Family of X st
( Y c= A & Y is finite & x = Intersect Y ) )

proof end;

uniqueness
for b₁, b₂ being Subset-Family of X st ( for x being Subset of X holds
( x in b₁ iff ex Y being Subset-Family of X st
( Y c= A & Y is finite & x = Intersect Y ) ) ) & ( for x being Subset of X holds
( x in b₂ iff ex Y being Subset-Family of X st
( Y c= A & Y is finite & x = Intersect Y ) ) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def3 defines FinMeetCl CANTOR_1:def 3 :
for X being set
for A, b₃ being Subset-Family of X holds
( b₃ = FinMeetCl A iff for x being Subset of X holds
( x in b₃ iff ex Y being Subset-Family of X st
( Y c= A & Y is finite & x = Intersect Y ) ) );

theorem Th4: :: CANTOR_1:4

for X being set
for A being Subset-Family of X holds A c= FinMeetCl A

proof end;

theorem Th5: :: CANTOR_1:5

for T being non empty TopSpace holds the topology of T = FinMeetCl the topology of T

proof end;

theorem Th6: :: CANTOR_1:6

for T being TopSpace holds the topology of T = UniCl the topology of T

proof end;

theorem Th7: :: CANTOR_1:7

for T being non empty TopSpace holds the topology of T = UniCl (FinMeetCl the topology of T)

proof end;

theorem Th8: :: CANTOR_1:8

for X being set
for A being Subset-Family of X holds X in FinMeetCl A

proof end;

theorem Th9: :: CANTOR_1:9

for X being set
for A, B being Subset-Family of X st A c= B holds
UniCl A c= UniCl B

proof end;

theorem Th10: :: CANTOR_1:10

for X being set
for R being non empty Subset-Family of (bool X)
for F being Subset-Family of X st F = { (Intersect x) where x is Element of R : verum } holds
Intersect F = Intersect (union R)

proof end;

theorem Th11: :: CANTOR_1:11

for X being set
for A being Subset-Family of X holds FinMeetCl A = FinMeetCl (FinMeetCl A)

proof end;

theorem :: CANTOR_1:12

for X being set
for A being Subset-Family of X
for a, b being set st a in FinMeetCl A & b in FinMeetCl A holds
a /\ b in FinMeetCl A

proof end;

theorem Th13: :: CANTOR_1:13

for X being set
for A being Subset-Family of X
for a, b being set st a c= FinMeetCl A & b c= FinMeetCl A holds
INTERSECTION (a,b) c= FinMeetCl A

proof end;

theorem Th14: :: CANTOR_1:14

for X being set
for A, B being Subset-Family of X st A c= B holds
FinMeetCl A c= FinMeetCl B

proof end;

registration

let X be set ;
let A be Subset-Family of X;

cluster FinMeetCl A -> non empty ;

coherence
not FinMeetCl A is empty by Th8;

end;

theorem Th15: :: CANTOR_1:15

for X being non empty set
for A being Subset-Family of X holds TopStruct(# X,(UniCl (FinMeetCl A)) #) is TopSpace-like

proof end;

definition

let X be TopStruct ;
let F be Subset-Family of X;

attr F is quasi_prebasis means :Def4: :: CANTOR_1:def 4
ex G being Basis of X st G c= FinMeetCl F;

end;

:: deftheorem Def4 defines quasi_prebasis CANTOR_1:def 4 :
for X being TopStruct
for F being Subset-Family of X holds
( F is quasi_prebasis iff ex G being Basis of X st G c= FinMeetCl F );

registration

let X be TopStruct ;

cluster the topology of X -> quasi_prebasis ;

coherence
the topology of X is quasi_prebasis by Th4;

cluster open quasi_basis -> quasi_prebasis for Element of bool (bool the carrier of X);

coherence
for b₁ being Basis of X holds b₁ is quasi_prebasis by Th4;

cluster open quasi_prebasis for Element of bool (bool the carrier of X);

existence
ex b₁ being Subset-Family of X st
( b₁ is open & b₁ is quasi_prebasis )

proof end;

end;

definition

let X be TopStruct ;
mode prebasis of X is open quasi_prebasis Subset-Family of X;

end;

theorem Th16: :: CANTOR_1:16

for X being non empty set
for Y being Subset-Family of X holds Y is Basis of TopStruct(# X,(UniCl Y) #) by Def2, Th1, TOPS_2:64;

theorem Th17: :: CANTOR_1:17

for T1, T2 being non empty strict TopSpace
for P being prebasis of T1 st the carrier of T1 = the carrier of T2 & P is prebasis of T2 holds
T1 = T2

proof end;

theorem Th18: :: CANTOR_1:18

for X being non empty set
for Y being Subset-Family of X holds Y is prebasis of TopStruct(# X,(UniCl (FinMeetCl Y)) #)

proof end;

definition

func the_Cantor_set -> non empty strict TopSpace means :: CANTOR_1:def 5
( the carrier of it = product (NAT --> {0,1}) & ex P being prebasis of it st
for X being Subset of (product (NAT --> {0,1})) holds
( X in P iff ex N, n being Nat st
for F being Element of product (NAT --> {0,1}) holds
( F in X iff F . N = n ) ) );

existence
ex b₁ being non empty strict TopSpace st
( the carrier of b₁ = product (NAT --> {0,1}) & ex P being prebasis of b₁ st
for X being Subset of (product (NAT --> {0,1})) holds
( X in P iff ex N, n being Nat st
for F being Element of product (NAT --> {0,1}) holds
( F in X iff F . N = n ) ) )

proof end;

uniqueness
for b₁, b₂ being non empty strict TopSpace st the carrier of b₁ = product (NAT --> {0,1}) & ex P being prebasis of b₁ st
for X being Subset of (product (NAT --> {0,1})) holds
( X in P iff ex N, n being Nat st
for F being Element of product (NAT --> {0,1}) holds
( F in X iff F . N = n ) ) & the carrier of b₂ = product (NAT --> {0,1}) & ex P being prebasis of b₂ st
for X being Subset of (product (NAT --> {0,1})) holds
( X in P iff ex N, n being Nat st
for F being Element of product (NAT --> {0,1}) holds
( F in X iff F . N = n ) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem defines the_Cantor_set CANTOR_1:def 5 :
for b₁ being non empty strict TopSpace holds
( b₁ = the_Cantor_set iff ( the carrier of b₁ = product (NAT --> {0,1}) & ex P being prebasis of b₁ st
for X being Subset of (product (NAT --> {0,1})) holds
( X in P iff ex N, n being Nat st
for F being Element of product (NAT --> {0,1}) holds
( F in X iff F . N = n ) ) ) );