:: The Sequential Closure Operator In Sequential and Frechet Spaces
:: by Bart{\l}omiej Skorulski
::
:: Received February 13, 1999
:: Copyright (c) 1999-2021 Association of Mizar Users

Lm1: for T being non empty TopSpace st ( for p being Point of T holds Cl {p} = {p} ) holds
T is T_1

proof end;

Lm2: for T being non empty TopSpace st not T is T_1 holds
ex x1, x2 being Point of T st
( x1 <> x2 & x2 in Cl {x1} )

proof end;

Lm3: for T being non empty TopSpace st not T is T_1 holds
ex x1, x2 being Point of T ex S being sequence of T st
( S = NAT --> x1 & x1 <> x2 & S is_convergent_to x2 )

proof end;

Lm4: for T being non empty TopSpace st T is T_2 holds
T is T_1

proof end;

theorem :: FRECHET2:1

for T being non empty 1-sorted
for S being sequence of T
for NS being increasing sequence of NAT holds S * NS is sequence of T ;

theorem :: FRECHET2:2

for RS being Real_Sequence st RS = id NAT holds
RS is increasing sequence of NAT ;

theorem Th3: :: FRECHET2:3

for T being non empty 1-sorted
for S being sequence of T
for A being Subset of T st ( for S1 being subsequence of S holds not rng S1 c= A ) holds
ex n being Element of NAT st
for m being Element of NAT st n <= m holds
not S . m in A

proof end;

theorem Th4: :: FRECHET2:4

for T being non empty 1-sorted
for S being sequence of T
for A, B being Subset of T st rng S c= A \/ B holds
ex S1 being subsequence of S st
( rng S1 c= A or rng S1 c= B )

proof end;

Lm5: for T being non empty TopSpace st T is first-countable holds
for x being Point of T ex B being Basis of x ex S being Function st
( dom S = NAT & rng S = B & ( for n, m being Element of NAT st m >= n holds
S . m c= S . n ) )

proof end;

theorem :: FRECHET2:5

for T being non empty TopSpace st ( for S being sequence of T
for x1, x2 being Point of T st x1 in Lim S & x2 in Lim S holds
x1 = x2 ) holds
T is T_1

proof end;

theorem Th6: :: FRECHET2:6

for T being non empty TopSpace st T is T_2 holds
for S being sequence of T
for x1, x2 being Point of T st x1 in Lim S & x2 in Lim S holds
x1 = x2

proof end;

theorem :: FRECHET2:7

for T being non empty TopSpace st T is first-countable holds
( T is T_2 iff for S being sequence of T
for x1, x2 being Point of T st x1 in Lim S & x2 in Lim S holds
x1 = x2 )

proof end;

theorem :: FRECHET2:8

for T being non empty TopStruct
for S being sequence of T st not S is convergent holds
Lim S = {}

proof end;

theorem Th9: :: FRECHET2:9

for T being non empty TopSpace
for A being Subset of T st A is closed holds
for S being sequence of T st rng S c= A holds
Lim S c= A by FRECHET:24;

theorem :: FRECHET2:10

for T being non empty TopStruct
for S being sequence of T
for x being Point of T st not S is_convergent_to x holds
ex S1 being subsequence of S st
for S2 being subsequence of S1 holds not S2 is_convergent_to x

proof end;

theorem Th11: :: FRECHET2:11

for T1, T2 being non empty TopSpace
for f being Function of T1,T2 st f is continuous holds
for S1 being sequence of T1
for S2 being sequence of T2 st S2 = f * S1 holds
f .: (Lim S1) c= Lim S2

proof end;

theorem :: FRECHET2:12

for T1, T2 being non empty TopSpace
for f being Function of T1,T2 st T1 is sequential holds
( f is continuous iff for S1 being sequence of T1
for S2 being sequence of T2 st S2 = f * S1 holds
f .: (Lim S1) c= Lim S2 )

proof end;

definition

let T be non empty TopStruct ;
let A be Subset of T;

func Cl_Seq A -> Subset of T means :Def1: :: FRECHET2:def 1
for x being Point of T holds
( x in it iff ex S being sequence of T st
( rng S c= A & x in Lim S ) );

existence
ex b₁ being Subset of T st
for x being Point of T holds
( x in b₁ iff ex S being sequence of T st
( rng S c= A & x in Lim S ) )

proof end;

uniqueness
for b₁, b₂ being Subset of T st ( for x being Point of T holds
( x in b₁ iff ex S being sequence of T st
( rng S c= A & x in Lim S ) ) ) & ( for x being Point of T holds
( x in b₂ iff ex S being sequence of T st
( rng S c= A & x in Lim S ) ) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def1 defines Cl_Seq FRECHET2:def 1 :
for T being non empty TopStruct
for A, b₃ being Subset of T holds
( b₃ = Cl_Seq A iff for x being Point of T holds
( x in b₃ iff ex S being sequence of T st
( rng S c= A & x in Lim S ) ) );

theorem Th13: :: FRECHET2:13

for T being non empty TopStruct
for A being Subset of T
for S being sequence of T
for x being Point of T st rng S c= A & x in Lim S holds
x in Cl A

proof end;

theorem Th14: :: FRECHET2:14

for T being non empty TopStruct
for A being Subset of T holds Cl_Seq A c= Cl A

proof end;

theorem Th15: :: FRECHET2:15

for T being non empty TopStruct
for S being sequence of T
for S1 being subsequence of S
for x being Point of T st S is_convergent_to x holds
S1 is_convergent_to x

proof end;

theorem Th16: :: FRECHET2:16

for T being non empty TopStruct
for S being sequence of T
for S1 being subsequence of S holds Lim S c= Lim S1

proof end;

theorem Th17: :: FRECHET2:17

for T being non empty TopStruct holds Cl_Seq ({} T) = {}

proof end;

theorem Th18: :: FRECHET2:18

for T being non empty TopStruct
for A being Subset of T holds A c= Cl_Seq A

proof end;

theorem Th19: :: FRECHET2:19

for T being non empty TopStruct
for A, B being Subset of T holds (Cl_Seq A) \/ (Cl_Seq B) = Cl_Seq (A \/ B)

proof end;

theorem Th20: :: FRECHET2:20

for T being non empty TopStruct holds
( T is Frechet iff for A being Subset of T holds Cl A = Cl_Seq A )

proof end;

theorem Th21: :: FRECHET2:21

for T being non empty TopSpace st T is Frechet holds
for A, B being Subset of T holds
( Cl_Seq ({} T) = {} & A c= Cl_Seq A & Cl_Seq (A \/ B) = (Cl_Seq A) \/ (Cl_Seq B) & Cl_Seq (Cl_Seq A) = Cl_Seq A )

proof end;

theorem Th22: :: FRECHET2:22

for T being non empty TopSpace st T is sequential & ( for A being Subset of T holds Cl_Seq (Cl_Seq A) = Cl_Seq A ) holds
T is Frechet

proof end;

theorem :: FRECHET2:23

for T being non empty TopSpace st T is sequential holds
( T is Frechet iff for A, B being Subset of T holds
( Cl_Seq ({} T) = {} & A c= Cl_Seq A & Cl_Seq (A \/ B) = (Cl_Seq A) \/ (Cl_Seq B) & Cl_Seq (Cl_Seq A) = Cl_Seq A ) ) by Th21, Th22;

definition

let T be non empty TopSpace;
let S be sequence of T;
assume A1: ex x being Point of T st Lim S = {x} ;

func lim S -> Point of T means :Def2: :: FRECHET2:def 2
S is_convergent_to it;

existence
ex b₁ being Point of T st S is_convergent_to b₁

proof end;

uniqueness
for b₁, b₂ being Point of T st S is_convergent_to b₁ & S is_convergent_to b₂ holds
b₁ = b₂

proof end;

end;

:: deftheorem Def2 defines lim FRECHET2:def 2 :
for T being non empty TopSpace
for S being sequence of T st ex x being Point of T st Lim S = {x} holds
for b₃ being Point of T holds
( b₃ = lim S iff S is_convergent_to b₃ );

theorem Th24: :: FRECHET2:24

for T being non empty TopSpace st T is T_2 holds
for S being sequence of T st S is convergent holds
ex x being Point of T st Lim S = {x}

proof end;

theorem Th25: :: FRECHET2:25

for T being non empty TopSpace st T is T_2 holds
for S being sequence of T
for x being Point of T holds
( S is_convergent_to x iff ( S is convergent & x = lim S ) )

proof end;

theorem :: FRECHET2:26

for M being MetrStruct
for S being sequence of M holds S is sequence of (TopSpaceMetr M)

proof end;

theorem :: FRECHET2:27

for M being non empty MetrStruct
for S being sequence of (TopSpaceMetr M) holds S is sequence of M by TOPMETR:12;

theorem Th28: :: FRECHET2:28

for M being non empty MetrSpace
for S being sequence of M
for x being Point of M
for S9 being sequence of (TopSpaceMetr M)
for x9 being Point of (TopSpaceMetr M) st S = S9 & x = x9 holds
( S is_convergent_in_metrspace_to x iff S9 is_convergent_to x9 )

proof end;

theorem :: FRECHET2:29

for M being non empty MetrSpace
for Sm being sequence of M
for St being sequence of (TopSpaceMetr M) st Sm = St holds
( Sm is convergent iff St is convergent )

proof end;

theorem :: FRECHET2:30

for M being non empty MetrSpace
for Sm being sequence of M
for St being sequence of (TopSpaceMetr M) st Sm = St & Sm is convergent holds
lim Sm = lim St

proof end;

definition

let T be TopStruct ;
let S be sequence of T;
let x be Point of T;

pred x is_a_cluster_point_of S means :: FRECHET2:def 3
for O being Subset of T
for n being Nat st O is open & x in O holds
ex m being Element of NAT st
( n <= m & S . m in O );

end;

:: deftheorem defines is_a_cluster_point_of FRECHET2:def 3 :
for T being TopStruct
for S being sequence of T
for x being Point of T holds
( x is_a_cluster_point_of S iff for O being Subset of T
for n being Nat st O is open & x in O holds
ex m being Element of NAT st
( n <= m & S . m in O ) );

theorem Th31: :: FRECHET2:31

for T being non empty TopStruct
for S being sequence of T
for x being Point of T st ex S1 being subsequence of S st S1 is_convergent_to x holds
x is_a_cluster_point_of S

proof end;

theorem :: FRECHET2:32

for T being non empty TopStruct
for S being sequence of T
for x being Point of T st S is_convergent_to x holds
x is_a_cluster_point_of S

proof end;

theorem Th33: :: FRECHET2:33

for T being non empty TopStruct
for S being sequence of T
for x being Point of T
for Y being Subset of T st Y = { y where y is Point of T : x in Cl {y} } & rng S c= Y holds
S is_convergent_to x

proof end;

theorem Th34: :: FRECHET2:34

for T being non empty TopStruct
for S being sequence of T
for x, y being Point of T st ( for n being Element of NAT holds
( S . n = y & S is_convergent_to x ) ) holds
x in Cl {y}

proof end;

theorem Th35: :: FRECHET2:35

for T being non empty TopStruct
for x being Point of T
for Y being Subset of T
for S being sequence of T st Y = { y where y is Point of T : x in Cl {y} } & rng S misses Y & S is_convergent_to x holds
ex S1 being subsequence of S st S1 is one-to-one

proof end;

theorem Th36: :: FRECHET2:36

for T being non empty TopStruct
for S1, S2 being sequence of T st rng S2 c= rng S1 & S2 is one-to-one holds
ex P being Permutation of NAT st S2 * P is subsequence of S1

proof end;

scheme :: FRECHET2:sch 1
PermSeq{ F₁() -> non empty 1-sorted , F₂() -> sequence of F₁(), F₃() -> Permutation of NAT, P₁[ set ] } :

ex n being Element of NAT st
for m being Element of NAT st n <= m holds
P₁[(F₂() * F₃()) . m]

provided

A1: ex n being Element of NAT st
for m being Element of NAT
for x being Point of F₁() st n <= m & x = F₂() . m holds
P₁[x]

proof end;

scheme :: FRECHET2:sch 2
PermSeq2{ F₁() -> non empty TopStruct , F₂() -> sequence of F₁(), F₃() -> Permutation of NAT, P₁[ set ] } :

ex n being Element of NAT st
for m being Element of NAT st n <= m holds
P₁[(F₂() * F₃()) . m]

provided

A1: ex n being Element of NAT st
for m being Element of NAT
for x being Point of F₁() st n <= m & x = F₂() . m holds
P₁[x]

proof end;

theorem Th37: :: FRECHET2:37

for T being non empty TopStruct
for S being sequence of T
for P being Permutation of NAT
for x being Point of T st S is_convergent_to x holds
S * P is_convergent_to x

proof end;

theorem :: FRECHET2:38

for n0 being Element of NAT ex NS being increasing sequence of NAT st
for n being Element of NAT holds NS . n = n + n0

proof end;

theorem Th39: :: FRECHET2:39

for T being non empty TopStruct
for S being sequence of T
for x being Point of T
for n0 being Nat st x is_a_cluster_point_of S holds
x is_a_cluster_point_of S ^\ n0

proof end;

theorem Th40: :: FRECHET2:40

for T being non empty TopStruct
for S being sequence of T
for x being Point of T st x is_a_cluster_point_of S holds
x in Cl (rng S)

proof end;

theorem :: FRECHET2:41

for T being non empty TopStruct st T is Frechet holds
for S being sequence of T
for x being Point of T st x is_a_cluster_point_of S holds
ex S1 being subsequence of S st S1 is_convergent_to x

proof end;

theorem :: FRECHET2:42

for T being non empty TopSpace st T is first-countable holds
for x being Point of T ex B being Basis of x ex S being Function st
( dom S = NAT & rng S = B & ( for n, m being Element of NAT st m >= n holds
S . m c= S . n ) ) by Lm5;

theorem :: FRECHET2:43

for T being non empty TopSpace st ( for p being Point of T holds Cl {p} = {p} ) holds
T is T_1 by Lm1;

theorem :: FRECHET2:44

for T being non empty TopSpace st T is T_2 holds
T is T_1 by Lm4;

theorem :: FRECHET2:45

for T being non empty TopSpace st not T is T_1 holds
ex x1, x2 being Point of T ex S being sequence of T st
( S = NAT --> x1 & x1 <> x2 & S is_convergent_to x2 ) by Lm3;