Research report: "Several have commented on the tight space count." potx

The sequential closure operator c∗on a topological space X, c possesses the following properties 1.1.. sequential, Fr´ echet, weakly first countable, countable tightness, closure operato

Notes on countable tightness spaces

Nguyen Van Dung(a)

Abstract In this paper we introduced the definition of countable closure operators

and use it to characterize countable tightness spaces Moreover, we consider the

gener-alizations of some results of W C Hong in[4],[5].

1 Introduction Let X be a set, P (X) the family of all subsets of X, S(X) the set of all se-quences of points in X, N the set of all natural numbers and R the set of all real numbers Let (X, c) be a topological space endowed with the closure operator c A function c∗ : P (X) −→ P (X)where c∗(A) = {x ∈ X|xn→ xfor some sequence {xn} ∈ S(A)}for every A ⊂ X is called asequential closure operator([5]) The sequential closure operator c∗on a topological space (X, c) possesses the following properties 1.1 Lemma ([5]) Let(X, c)be a topological space with the closure operatorc

(1) c∗(φ) = φ,

(2) A ⊂ c∗(A) ⊂ c(A)for everyA ⊂ X,

(3) If A ⊂ B ⊂ X, thenc∗(A) ⊂ c∗(B),

(4) c∗(A ∪ B) = c∗(A) ∪ c∗(B)for everyA, B ⊂ X

In other way, c∗ possesses the Kuratowski closure axoms except for idempo-tent So, it need not be a topological closure operator on the set X Recall the following definitions in a topological space (X, c) endowed with the closure oper-ator c

1.2 Definition Let X be a topological space

(1) X is called a sequential space ([1]) if for each subset A of X, A is closed provided that if {xn} ∈ S(A)and {xn}converges to x, then x ∈ A

(2) X is called a Frechet space ([1]) if for each subset A of X and any point

x ∈ c(A), there exists some sequence {xn} ⊂ Asuch that {xn} converges

to x

1 - 2000 Mathematics Subject Classification 54A20, 54D35, 54D55, 54D80, 55E25.

- Keywords sequential, Fr´ echet, weakly first countable, countable tightness, closure operator, sequential closure operator, countable closure operator.

- Received 10/01/2006, in revised form 25/02/2006.

(3) X is called acountable tightness space(or aspaceXhas countable tightness) ([1]) if for each subset A of X, A is closed provided that c(C) ⊂ A for every countable subset C of A

(4) X is called aweakly first countable space([5]) if for each x ∈ X, there exists

a family {B(x, n)|n ∈ N} of subsets of X such that the following conditions are satisfied

(a) x ∈ B(x, n + 1) ⊂ B(x, n)for all n ∈ N,

(b) a subset U of X is open in X if and only if for every x ∈ U there exists

an n ∈ N such that B(x, n) ⊂ U

(5) X is called a hereditarily sequential space ([1]) if every subspace of X is sequential

It is well known that ([1], [5])

(1) first countable ⇒ Frechet ⇒ sequential ⇒ countable tightness

(2) Frechet ⇔ hereditarily sequential

The following properties have been given in [5]

(1) (C∗): for each countable subset A of X, c∗(A) = c(A)

(2) (C∗∗): for each subset A of X which is not closed and each x ∈ c(A) \ A, there exists a subset B of A such that x ∈ c(B) and c(B) \ {x} ⊂ A (3) (C∗∗∗): for each countable subset A of X which is not closed and each x ∈ c(A)\A, there exists a subset B of A such that x ∈ c(B) and c(B)\{x} ⊂ A The sequential closure operator is used to characterize Frechet spaces and give conditions that a sequential space be a Frechet space ([4], [5]) We note that the definition of countable tightness is analogous to that of sequential, in the sense that if you replace ``countable subsetC'' by ``sequenceC'' and ``c(C)'' by ``set

of limit points ofC'' Hence, from the fact that a space is Frechet if and only if

``c∗ = c'' ([4]) and following results

1.3 Lemma (1) ([5], Theorem 2) A sequential space is Frechet if and only if it

satisfies the property(C∗)

(2) ([5], Theorem 4) Every sequential space satisfying the property(C∗∗∗) satis-fies the property(C∗)

(3) ([5], Corollary 6) Every Hausdorff Frechet space satisfies the property(C∗∗)

a question naturally rises that what the analogous results for the countable tight-ness spaces are

In this paper we introduce the countable closure operator c∗ and research some properties of countable tightness spaces by using this notion and the se-quential closure operator And then we obtain generalizations of the results of W

C Hong in [4], [5]

Standard notations, not explained below, is the same in [4], [5]

2 The main results Let (X, c) be a topological space with the closure operator c A function c∗ :

P (X) −→ P (X)defined by for each A ⊂ X,

c∗(A) = {x ∈ X|x ∈ c(C)for some countable subset C of A },

is called acountable closure operatoron the space (X, c)

2.1 Theorem Let(X, c)be a topological space with the closure operatorc

(1) c∗(φ) = φ,

(2) If A ⊂ B, thenc∗(A) ⊂ c∗(B)for everyA, B ⊂ X,

(3) c∗(A ∪ B) = c∗(A) ∪ c∗(B)for everyA, B ⊂ X,

(4) c∗(A) = c(A)for every countable subsetAofX,

(5) A ⊂ c∗(A) ⊂ c∗(A) ⊂ c(A)for everyA ⊂ X,

(6) c∗(A) = (c∗)∗(A)for everyA ⊂ X, where(c∗)∗(A) = c∗(c∗(A))

Proof It is easy to prove (1), (2), (4), (5)

(3) Since A ⊂ A ∪ B and B ⊂ A ∪ B, c∗(A) ∪ c∗(B) ⊂ c∗(A ∪ B) Suppose that x ∈ c∗(A ∪ B), then x ∈ c(C) for some countable subset C of A ∪ B We have c(C) = c((C ∩ A) ∪ (C ∩ B)) = c(C ∩ A) ∪ c(C ∩ B), x ∈ c(C ∩ A)or x ∈ c(C ∩ B) Obviously, C ∩ A and C ∩ B are countable subsets of A and B respectively So

x ∈ c∗(A) ∪ c∗(B)

(6) Obviously, c∗(A) ⊂ (c∗)∗(A) Suppose that x ∈ (c∗)∗(A), then x ∈ c(C) for some countable subset C of c∗(A) Assume that C = {xi : i ∈ N} ⊂ c∗(A) Since for every i ∈ N, xi ∈ c(Ci)for some countable subset Ci of A and [

i∈N

Ci is

countable in A, and we have C ⊂ [

i∈N

c(Ci) ⊂ c [

i∈N

Ci

! Hence by properties of

closure we get x ∈ c(C) ⊂ c [

i∈N

Ci

! , where [

i∈N

Ci is countable in A It follows that (c∗)∗(A) ⊂ c∗(A) Thus c∗(A) = (c∗)∗(A)  2.2 Remark (1) For every subset A of a Frechet space X we have c∗(A) =

c∗(A) = c(A)

(2) The property (C∗)is equivalent to following property (C∗)0,

(C∗)0: for each countable subset A of X, c∗(A) = c∗(A)

(3) Each countable closure operator is a topological closure operator

Next, we give a condition for a space to have the countable tightness

2.3 Theorem Let(X, c) be a topological space with the closure operator c Then

(X, c)is a countable tightness space if and only ifc∗(A) = c(A)for everyA ⊂ X

Proof Suppose that (X, c) has the countable tightness and A ⊂ X It is sufficient

to prove that c(A) ⊂ c∗(A) We have A ⊂ c∗(A), so we only prove that c∗(A) is closed Let C be an arbitrary countable subset of c∗(A) and x ∈ c(C) Since for each y ∈ C ⊂ c∗(A), there exists a countable subset Cy of A such that y ∈ c(Cy) Denote D = Sy∈CCy, then D is a countable subset of A Let U be an arbitrary neighbourhood of x then U ∩ C 6= φ Hence, there exists y ∈ U and y ∈ C Since

y ∈ c(Cy), we have U ∩ Cy 6= φ So U ∩ D 6= φ This shows that x ∈ c(D) Hence, c(C) ⊂ c∗(A) From the fact that (X, c) has countable tightness, c∗(A)is closed Conversely, suppose that c∗(A) = c(A) for every A ⊂ X Let B ⊂ X and c(C) ⊂ B for every countable subset C of B Then c∗(B) ⊂ B Hence, c(B) ⊂ B

It follows that B is closed Hence, (X, c) has the countable tightness  From this Theorem we have the following

2.4 Corollary For a countable tightness space X, the following statements are equivalent

(1) X is Frechet,

(2) X is hereditarily sequential,

(3) c∗(A) = c∗(A) = c(A)for everyA ⊂ X

2.5 Corollary If (X, c) is a topological space with the closure operator c, then

(X, c∗)is a countable tightness space

Proof From Theorem 2.1 we have that (X, c∗) is topological space with closure operator c∗ which satisfies c∗(c∗(A)) = c∗(A) for every A ∈ P (X) It follows that (X, c∗)has the countable tightness by Theorem 2.3  2.6 Corollary A countable tightness space(X, c)is Frechet if and only if it satisfies the property(C∗)

Proof Suppose that a countable tightness space (X, c) satisfies the property (C∗)

By Theorem 2.3, c∗(A) = c(A)for every subset A of X

Next, we prove that c∗(A) = c∗(A)for every subset A of X Obviously, c∗(A) ⊂

c∗(A) For each x ∈ c∗(A), there exists a countable subset C of A such that x ∈ c(C) Since (X, c) satisfies the property (C∗), c(C) = c∗(C) ⊂ c∗(A) So, x ∈ c∗(A)

It follows that c∗(A) = c∗(A)

Thus, c∗(A) = c∗(A) = c(A) for every subset A of X Hence, X is Frechet by Corollary 2.4 The converse is clearly  2.7 Corollary For a countable tightness space X, the following statements are equivalent

(1) X is Frechet,

(2) X is hereditarily sequential,

(3) X satisfies(C∗),

(4) for each countable subsetAofX,c∗(A) = c∗(A) = c(A)

Proof (1) ⇔ (2) is well known.

(1) ⇒ (3) By Lemma 1.3.(1)

(3) ⇒ (4) By Theorem 2.1.(4)

Recall that a space X is called a k-space if A ∈ P (X) is open provided that

A ∩ K is open in K for every compact subset K of X ([7]) It is well known that each Hausdorff sequential space is a k-space ([1]) Hence, the class of Hausdorff

k-spaces and the class of Hausdorff countable tightness spaces both contain the class of Hausdorff sequential spaces But they are distinct by following example 2.8 Example Let X = {0}∪(

∞

[

i=1

Xi)where Xi =1i ∪(

∞

[

j=i 2

n1

i+

1 j

o )for every i =

1, 2, · · · and topology in X is generated by the neighbourhood system {B(x)|x ∈ X}as following

If x = 1

i +1j then B(x) = {x}

If x = 1

i then B(x) consists all sets of the form 1

i ∪ (

∞

[

j=k

n1

i +

1 j

o ), k =

i2, i2+ 1, · · ·

If x = 0 then B(0) consists all sets which obtain from X by removing a finite number of Xi's and a finite number of points of the form 1

i +1j in all remaining

Xi's

Then Y = X \n1,1

2, · · ·

o

is not a k-space ([1], Examples 3.3.24) but Haus-dorff Because Y is countable, Y has countable tightness by Theorem 2.1.(4) and Theorem 2.3

The following results are proved in [4], [5]

2.9 Lemma (1) ([5], Corollary 5) Every sequential space satisfying the

prop-erty(C∗∗∗)is Frechet

(2) ([4], Theorem 2.10)Every weakly first countable space satisfying the property

(C∗∗)is first countable

2.10 Remark (1) Every countable space has countable tightness by

Theo-rem 2.1.(4) and TheoTheo-rem 2.3

(2) Since each sequential space is a countable tightness space, Corollary 2.6

is a generalization of Theorem 2 in [5]

(3) ([5], Example (4)) The space X = {a, b, c} with the topology {φ, X, {a}} is Frechet and hence it is sequential X satisfies the property (C∗) but it does not satisfy the property (C∗∗∗) It follows that the converse of Lemma 2.9.(1) is not true

It is well known that each weakly first countable space is a sequential space, and each sequential space is a countable tightness space The following exam-ples prove that Lemma 1.3.(2) and Lemma 2.9.(1) can not generalize to countable tightness spaces and Lemma 2.9.(2) can not generalize to sequential spaces 2.11 Example The space Y in Example 2.8 is a countable tightness space which

is not a k-space Thus, Y is neither a Frechet space nor a sequential space We prove that Y satisfies the property (C∗∗) Suppose that A is an arbitrary countable subset of Y which is not closed Then {i|A ∩ Xi is infinite } is infinite and 0 /∈ A ([1], Example 1.6.19) We have c(A) = A ∪ {0}, so c(A) \ {0} ⊂ A Hence, the property (C∗∗)is satisfied

Since Y satisfies the property (C∗∗), Y satisfies the property (C∗∗∗) Note that

Y is not Frechet, Y is not satisfied the property (C∗)by Corollary 2.6

2.12 Example Let X = R with the usual topology and Y = (R\N)∪{∞} Assign

to any point x ∈ X the point

f (x) =

(

x if x ∈ R \ N,

∞ if x ∈ N

and consider on Y the topology generated by the family of closed sets

{A ⊂ Y |f−1(A)is closed in X}

Then, following facts are easily proved ([1], Example 1.6.18)

(1) Let A be a subset of Y Then A is open and ∞ ∈ A if and only if there exists

a subset U such that U is open in X and N ⊂ U and A = (U \ N) ∪ {∞} (2) Let A be a subset of Y Then A is open and ∞ /∈ A if and only if A = B where B is open in X and B ∩ N = φ

(3) Y is not first countable

(4) Y is Frechet and Hausdorff

Since every Hausdorff Frechet space satisfies the property (C∗∗), Y satisfies the property (C∗∗) Hence Y is the sequential space which satisfies the property (C∗∗)is not first countable

By the same of the property (C∗∗),we give the following property

(C∗∗): for each subset A of X, which is not closed in X and each x ∈ c(A) \ A, there exists a subset B of A such that x ∈ c∗(B)and c∗(B) \ {x} ⊂ A

It is clearly that for a countable tightness space X, (C∗∗)and (C∗∗)are equiv-alent

2.13 Theorem A spaceXsatisfying for each non-closed subsetAofX, there exists

a subsetB ofAsuch thatc∗(B) 6⊂ Aif and only if it has countable tightness

Proof If X has countable tightness and A is not closed in X There exists a count-able subset B of A such that c(B) 6⊂ A Since B is countcount-able, c(B) = c∗(B) We have B ⊂ A and c∗(B) 6⊂ A

Conversely, let A ⊂ X and c(C) ⊂ A for every countable subset C of A If

A is not closed, there exists a subset B of A such that c∗(B) 6⊂ A So there is a countable set C ⊂ B ⊂ A such that c(C) 6⊂ A This is a contradiction Thus A is closed So X has countable tightness  2.14 Corollary A space satisfying the property(C∗∗)has countable tightness

Proof Suppose that X satisfies the proprerty (C∗∗)and A is not closed in X There exists x ∈ c(A) \ A and a subset B of A such that x ∈ c∗(B)and c∗(B) \ {x} ⊂ A

So c∗(B) 6⊂ A Thus X has countable tightness by Theorem 2.13  2.15 Remark The space X = {a, b, c} with topology {φ, {a}, X} is Frechet and satisfies the property (C∗)but it does not satisfy the property (C∗∗)([5]) Hence X

is a countable tightness space which does not satisfy the property (C∗∗) It follows that the converse of Corollary 2.14 is not true

References

[1] R Engelking, General topology, PWN-Polish Scientific Publishers, Warszawa 1977.

[2] S P Franklin, Spaces in which sequences suffice, Fund Math., 57 (1965), 108 - 115.

[3] S P Franklin, Spaces in which sequences suffice II, Fund Math., 61 (1967), 51 - 56.

[4] W C Hong, Notes on Fr´ echet, Internat J Math & Math Sci., 22 (3) (1999), 659 - 665 [5] W C Hong, On sequential spaces and a related class of spaces, Kyungpook Math J., 40 (2000), 149 - 155.

[6] B Skorulski, First countable, Sequential and Fr´ echet spaces, J of Formalized Math., Vol 10 (2003), 1 - 5.

[7] Y Tanaka, Theory of k-networks II, Q & A in Topology, Vol 19 (2001), 27 - 46.

tóm tắt vài nhận xét về không gian có tính chặt đếm được

Trong bài báo này chúng tôi giới thiệu toán tử bao đóng đếm được và dùng nó

để đặc trưng các không gian có tính chặt đếm được Tiếp đó chứng tôi đưa ra một vài mở rộng của các kết quả của W C Hong trong [4], [5]

(a) Department of Mathematics, DongThap Pedagogical University