:: Components and Basis of Topological Spaces
:: by Robert Milewski
::
:: Received June 22, 1999
:: Copyright (c) 1999-2021 Association of Mizar Users

scheme :: YELLOW15:sch 1
SeqLambda1C{ F₁() -> Nat, F₂() -> non empty set , P₁[ object ], F₃( object ) -> set , F₄( object ) -> set } :

ex p being FinSequence of F₂() st
( len p = F₁() & ( for i being Nat st i in Seg F₁() holds
( ( P₁[i] implies p . i = F₃(i) ) & ( P₁[i] implies p . i = F₄(i) ) ) ) )

provided

A1: for i being Nat st i in Seg F₁() holds
( ( P₁[i] implies F₃(i) in F₂() ) & ( P₁[i] implies F₄(i) in F₂() ) )

proof end;

definition

let X be set ;
let p be FinSequence of bool X;
let q be FinSequence of BOOLEAN ;

func MergeSequence (p,q) -> FinSequence of bool X means :Def1: :: YELLOW15:def 1
( len it = len p & ( for i being Nat st i in dom p holds
it . i = IFEQ ((q . i),TRUE,(p . i),(X \ (p . i))) ) );

existence
ex b₁ being FinSequence of bool X st
( len b₁ = len p & ( for i being Nat st i in dom p holds
b₁ . i = IFEQ ((q . i),TRUE,(p . i),(X \ (p . i))) ) )

proof end;

uniqueness
for b₁, b₂ being FinSequence of bool X st len b₁ = len p & ( for i being Nat st i in dom p holds
b₁ . i = IFEQ ((q . i),TRUE,(p . i),(X \ (p . i))) ) & len b₂ = len p & ( for i being Nat st i in dom p holds
b₂ . i = IFEQ ((q . i),TRUE,(p . i),(X \ (p . i))) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def1 defines MergeSequence YELLOW15:def 1 :
for X being set
for p being FinSequence of bool X
for q being FinSequence of BOOLEAN
for b₄ being FinSequence of bool X holds
( b₄ = MergeSequence (p,q) iff ( len b₄ = len p & ( for i being Nat st i in dom p holds
b₄ . i = IFEQ ((q . i),TRUE,(p . i),(X \ (p . i))) ) ) );

theorem :: YELLOW15:1

canceled;

theorem :: YELLOW15:2

canceled;

theorem :: YELLOW15:3

canceled;

::$CT 3

theorem Th1: :: YELLOW15:4

for X being set
for p being FinSequence of bool X
for q being FinSequence of BOOLEAN holds dom (MergeSequence (p,q)) = dom p

proof end;

theorem Th2: :: YELLOW15:5

for X being set
for p being FinSequence of bool X
for q being FinSequence of BOOLEAN
for i being Nat st q . i = TRUE holds
(MergeSequence (p,q)) . i = p . i

proof end;

theorem Th3: :: YELLOW15:6

for X being set
for p being FinSequence of bool X
for q being FinSequence of BOOLEAN
for i being Nat st i in dom p & q . i = FALSE holds
(MergeSequence (p,q)) . i = X \ (p . i)

proof end;

theorem :: YELLOW15:7

for X being set
for q being FinSequence of BOOLEAN holds len (MergeSequence ((<*> (bool X)),q)) = 0

proof end;

theorem Th5: :: YELLOW15:8

for X being set
for q being FinSequence of BOOLEAN holds MergeSequence ((<*> (bool X)),q) = <*> (bool X)

proof end;

theorem :: YELLOW15:9

for X being set
for x being Subset of X
for q being FinSequence of BOOLEAN holds len (MergeSequence (<*x*>,q)) = 1

proof end;

theorem :: YELLOW15:10

for X being set
for x being Subset of X
for q being FinSequence of BOOLEAN holds
( ( q . 1 = TRUE implies (MergeSequence (<*x*>,q)) . 1 = x ) & ( q . 1 = FALSE implies (MergeSequence (<*x*>,q)) . 1 = X \ x ) )

proof end;

theorem :: YELLOW15:11

for X being set
for x, y being Subset of X
for q being FinSequence of BOOLEAN holds len (MergeSequence (<*x,y*>,q)) = 2

proof end;

theorem :: YELLOW15:12

for X being set
for x, y being Subset of X
for q being FinSequence of BOOLEAN holds
( ( q . 1 = TRUE implies (MergeSequence (<*x,y*>,q)) . 1 = x ) & ( q . 1 = FALSE implies (MergeSequence (<*x,y*>,q)) . 1 = X \ x ) & ( q . 2 = TRUE implies (MergeSequence (<*x,y*>,q)) . 2 = y ) & ( q . 2 = FALSE implies (MergeSequence (<*x,y*>,q)) . 2 = X \ y ) )

proof end;

theorem :: YELLOW15:13

for X being set
for x, y, z being Subset of X
for q being FinSequence of BOOLEAN holds len (MergeSequence (<*x,y,z*>,q)) = 3

proof end;

theorem :: YELLOW15:14

for X being set
for x, y, z being Subset of X
for q being FinSequence of BOOLEAN holds
( ( q . 1 = TRUE implies (MergeSequence (<*x,y,z*>,q)) . 1 = x ) & ( q . 1 = FALSE implies (MergeSequence (<*x,y,z*>,q)) . 1 = X \ x ) & ( q . 2 = TRUE implies (MergeSequence (<*x,y,z*>,q)) . 2 = y ) & ( q . 2 = FALSE implies (MergeSequence (<*x,y,z*>,q)) . 2 = X \ y ) & ( q . 3 = TRUE implies (MergeSequence (<*x,y,z*>,q)) . 3 = z ) & ( q . 3 = FALSE implies (MergeSequence (<*x,y,z*>,q)) . 3 = X \ z ) )

proof end;

theorem Th12: :: YELLOW15:15

for X being set
for p being FinSequence of bool X holds { (Intersect (rng (MergeSequence (p,q)))) where q is FinSequence of BOOLEAN : len q = len p } is Subset-Family of X

proof end;

registration

cluster -> boolean-valued for FinSequence of BOOLEAN ;

coherence
for b₁ being FinSequence of BOOLEAN holds b₁ is boolean-valued

proof end;

end;

definition

let X be set ;
let Y be finite Subset-Family of X;

func Components Y -> Subset-Family of X means :Def2: :: YELLOW15:def 2
ex p being FinSequence of bool X st
( len p = card Y & rng p = Y & it = { (Intersect (rng (MergeSequence (p,q)))) where q is FinSequence of BOOLEAN : len q = len p } );

existence
ex b₁ being Subset-Family of X ex p being FinSequence of bool X st
( len p = card Y & rng p = Y & b₁ = { (Intersect (rng (MergeSequence (p,q)))) where q is FinSequence of BOOLEAN : len q = len p } )

proof end;

uniqueness
for b₁, b₂ being Subset-Family of X st ex p being FinSequence of bool X st
( len p = card Y & rng p = Y & b₁ = { (Intersect (rng (MergeSequence (p,q)))) where q is FinSequence of BOOLEAN : len q = len p } ) & ex p being FinSequence of bool X st
( len p = card Y & rng p = Y & b₂ = { (Intersect (rng (MergeSequence (p,q)))) where q is FinSequence of BOOLEAN : len q = len p } ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def2 defines Components YELLOW15:def 2 :
for X being set
for Y being finite Subset-Family of X
for b₃ being Subset-Family of X holds
( b₃ = Components Y iff ex p being FinSequence of bool X st
( len p = card Y & rng p = Y & b₃ = { (Intersect (rng (MergeSequence (p,q)))) where q is FinSequence of BOOLEAN : len q = len p } ) );

registration

let X be set ;
let Y be finite Subset-Family of X;

cluster Components Y -> finite ;

coherence
Components Y is finite

proof end;

end;

theorem Th13: :: YELLOW15:16

for X being set
for Y being empty Subset-Family of X holds Components Y = {X}

proof end;

theorem :: YELLOW15:17

for X being set
for Y, Z being finite Subset-Family of X st Z c= Y holds
Components Y is_finer_than Components Z

proof end;

theorem Th15: :: YELLOW15:18

for X being set
for Y being finite Subset-Family of X holds union (Components Y) = X

proof end;

theorem Th16: :: YELLOW15:19

for X being set
for Y being finite Subset-Family of X
for A, B being set st A in Components Y & B in Components Y & A <> B holds
A misses B

proof end;

definition

let X be set ;
let Y be finite Subset-Family of X;

attr Y is in_general_position means :: YELLOW15:def 3
not {} in Components Y;

end;

:: deftheorem defines in_general_position YELLOW15:def 3 :
for X being set
for Y being finite Subset-Family of X holds
( Y is in_general_position iff not {} in Components Y );

theorem :: YELLOW15:20

for X being set
for Y, Z being finite Subset-Family of X st Z is in_general_position & Y c= Z holds
Y is in_general_position

proof end;

theorem :: YELLOW15:21

for X being non empty set
for Y being empty Subset-Family of X holds Y is in_general_position

proof end;

theorem :: YELLOW15:22

for X being non empty set
for Y being finite Subset-Family of X st Y is in_general_position holds
Components Y is a_partition of X

proof end;

theorem Th20: :: YELLOW15:23

for L being non empty RelStr holds
( [#] L is infs-closed & [#] L is sups-closed )

proof end;

theorem Th21: :: YELLOW15:24

for L being non empty RelStr holds
( [#] L is with_bottom & [#] L is with_top )

proof end;

registration

let L be non empty RelStr ;

cluster [#] L -> infs-closed sups-closed with_bottom with_top ;

coherence
( [#] L is infs-closed & [#] L is sups-closed & [#] L is with_bottom & [#] L is with_top ) by Th20, Th21;

end;

theorem :: YELLOW15:25

for L being continuous sup-Semilattice holds [#] L is CLbasis of L

proof end;

theorem :: YELLOW15:26

for L being non empty up-complete Poset st L is finite holds
the carrier of L = the carrier of (CompactSublatt L)

proof end;

theorem :: YELLOW15:27

for L being lower-bounded sup-Semilattice
for B being Subset of L st B is infinite holds
card B = card (finsups B)

proof end;

theorem :: YELLOW15:28

for T being non empty T_0 TopSpace holds card the carrier of T c= card the topology of T

proof end;

theorem Th26: :: YELLOW15:29

for T being TopStruct
for X being Subset of T st X is open holds
for B being finite Subset-Family of T st B is Basis of T holds
for Y being set holds
( not Y in Components B or X misses Y or Y c= X )

proof end;

theorem :: YELLOW15:30

for T being T_0 TopSpace st T is infinite holds
for B being Basis of T holds B is infinite

proof end;

theorem :: YELLOW15:31

for T being non empty TopSpace st T is finite holds
for B being Basis of T
for x being Element of T holds meet { A where A is Element of the topology of T : x in A } in B

proof end;