A STUDY ON SEQUENTIAL SPACES IN TOPOLOGICAL AND IDEAL TOPOLOGICAL SPACES

Synopsis submitted to Madurai Kamaraj University

in partial fulfilment of the requirements for the award of the degree of

Doctor of Philosophy in by P. Vijaya Shanthi (Registration No. F 9832)

under the Guidance of Dr. V. Renuka Devi Assistant Professor

Centre for Research and Post Graduate Studies in Mathematics Ayya Nadar Janaki Ammal College (Autonomous, Affiliated to Madurai Kamaraj University, Re-accredited (3rd cycle) with ‘A’ Grade (CGPA 3.67 out of 4) by NAAC, recognized as College of Excellence & Mentor College by UGC, STAR College by DBT and Ranked 51st at National Level in NIRF 2019) Sivakasi - 626 124 Tamil Nadu India March - 2020 SYNOPSIS

The thesis entitled “A study on sequential spaces in topological and ideal topological spaces” consists of five chapters.

In this thesis, we discuss about statistically convergence of , se- quentially I-convergence structure and generalized convergence of a in topological and ideal topological spaces. First chapter entitled “Preliminaries” consisting of basic definitions and some results which are used in this thesis.

The idea of statistical convergence for real sequence was introduced by Fast

[5], Steinhaus [13]. Cakalh [1] gave characterizations of statistical convergence for bounded sequences. Recently, Pehlivan and Mamedov [12] proved that all optimal paths have the same unique statistical cluster point which is also a statistical point. Some basic properties of statistical convergence were proved by Connor and

Kline [3]. In general, statistical convergent sequences statisfy many of the properties of ordinary convergent sequences in spaces [8].

Fridy [6] introduced the notions of statistical limit and statistical cluster points and further these concepts were studied by Kostyrko, Macaj, Sal´atandˇ

Strauch [8]. Also, they proved that the set of all statistical cluster points is a closed set but the set of all statistical limit points is neither open nor closed.

In the second chapter entitled, “Statistical version of compact spaces”, we introduce the concepts of countably s-Fr´echet-Urysohn, s-sequential compact and maximal s-sequential compact spaces. Then we prove the main theorem on A study on sequential spaces in topological and ideal topological spaces countably s-Fr´echet-Urysohn spaces using these concepts. In the second section of this chapter, we extend the study on statistically sequentially countably compact spaces and give relation between statistically sequentially compact and statistically sequentially countably compact spaces.

Definition 1. Let (X, τ) be a . A [·]s−seq of the power

s set P(X) to itself defined by for each subset A of X,[A]s−seq = {x ∈ X | (xn) −→ x in (X, τ) for some sequence (xn) of points in A} is called the s-sequential operator on X.

The following Theorem 2 gives some properties of s-sequential closure operator on a topological space.

Theorem 2. Let A and B be any two subsets of a topological space (X, τ). Then the following hold.

(a) [∅]s−seq = ∅.

(b) A ⊂ [A]s−seq.

(c) [A]s−seq ⊂ cl(A).

(d) A ⊂ B ⇒ [A]s−seq ⊂ [B]s−seq.

(e) [A]s−seq ⊂ [[A]s−seq]s−seq.

Lemma 3. If (X, τ) is a countably s-Fr´echet-Urysohnspace, then [·]s−seq is a Ku- ratowski’s closure operator on X and the family τs−seq = {X \ [A]s−seq | A ⊂ X} is a on the set X.

2 A study on sequential spaces in topological and ideal topological spaces

Theorem 4. Let (X, τ) be a countably s-Fr´echet-Urysohnspace. If τs−seq = {X \

[A]s−seq | A ⊂ X}, then (X, τs−seq) is a s-Fr´echet-Urysohnspace. Moreover, for

s each sequence (xn) of points in X and each p ∈ X, (xn) −→ p in (X, τ) if and only if

s (xn) −→ p in (X, τs−seq).

Theorem 5. Let (X, τ) be a non-s-sequentially compact space, X∗ = X ∪ {∞} with

∞ ∈/ X and τ ∗ = τ ∪ {U ⊂ X∗ | ∞ ∈ U, X − U is a closed, s-sequentially compact subset of (X, τ) } Then (X∗, τ ∗) is a s-sequentially compact space with unique s- sequential limits and satisfies the following property: for each sequence (xn) of points

s s ∗ ∗ in X and each p ∈ X, (xn) −→ p in (X, τ) if and only if (xn) −→ p in (X , τ ).

Theorem 6. Let (X, τ) be a countably s-Fr´echetUrysohn space. Then the following hold.

(a) (X, τ) is a maximal s-sequentially compact space if and only if (X, τ) is a s-

Fr´echet-Urysohnspace.

(b) Suppose X is a non s-sequentially compact space. Then the one-point-s-sequential compactification (X∗, τ ∗) of (X, τ) is maximal s-sequentially compact if and only if

(X, τ) is a s-Fr´echet-Urysohnspace.

The following Theorem 7 shows that every ss-countably compact space is s- sequentially compact in a s-.

Theorem 7. Every s-sequential, ss-countably compact space is a s-sequentially com- pact space.

3 A study on sequential spaces in topological and ideal topological spaces

The following Theorem 8 gives a necessary and sufficient condition for a contin- uous map having s-sequential space as its domain.

Theorem 8. Let (X, τ) be a s-sequential space and (Y, σ) be a topological space.

s s Then f : X → Y is continuous if and only if (xn) −→ x implies that (f(xn)) −→ f(x).

Levine [9] introduced the concept of generalized closed sets (briefly, g-closed set) in a topological space. Caldas and Jafari [2] introduced new type of convergence in terms of g-open sets. Also, they studied sequentially g-closed sets and sequentially g-continuous map by utilizing g-open sets.

In the third chapter entitled, “On sequential g-convergence spaces”, we introduce the concepts such as sequentially g-open sets, g-sequential spaces, g-

Fr´echet spaces and sequentially g-quotient map using g-convergence sequences. Then we discuss the properties of sequentially g-closed sets and sequentially g-continuous maps. It follows that every g-Fr´echet-Urysohn space is a g-sequential space. Next, we characterize and study the properties of g-sequential spaces, g-Fr´echet spaces and sequentially g-quotient map.

Definition 9. Let (X, τ) be a topological space, A ⊂ X and let S[A] be the set of all

sequences in A. Then the sequential g-closure of A, denoted by [A]gseq , is defined as

[A]gseq = {x ∈ X | x = glim xn and (xn) ∈ S[A] ∩ cg(X)}

cg(X) denote the set of all g-convergent sequences in X.

The following Theorem 10 gives the properties of sequential g-closure operator.

4 A study on sequential spaces in topological and ideal topological spaces

Theorem 10. Let A and B be subsets of a topological space (X, τ). Then the following hold.

(a) [∅]gseq = ∅.

(b) A ⊂ [A]gseq .

(c) [A]gseq ⊂ cl(A).

(d) A ⊂ B ⇒ [A]gseq ⊂ [B]gseq .

(e) [A]gseq ∪ [B]gseq = [A ∪ B]gseq .

(f) [A]gseq ⊂ [[A]gseq ]gseq .

We introduce the definition of g-sequential and g-Fr´echet spaces. Also, we inves- tigate some properties of g-sequential and g-Fr´echet spaces. We shall see that each g-Fr´echet space is a g-sequential space and each g-Fr´echet space is a Fr´echet space.

However, the converse implications of the above statements are not true in general.

Definition 11. A topological space (X, τ) is said to be g-sequential if any subset

A of X with [A]gseq ⊂ A is closed in X, that is, every sequentially g-closed set in X is a closed set.

Definition 12. A topological space (X, τ) is said to be g-Fr´echet if cl(A) ⊂ [A]gseq for each A ⊂ X.

Theorem 13. Let (X, τ) be a topological space. Then the following hold.

(a) If X is a g-sequential space, then X is a sequential space.

(b) If X is a g-Fr´echetspace, then X is a Fr´echetspace.

5 A study on sequential spaces in topological and ideal topological spaces

(c) If X is a g-Fr´echetspace, then X is a g-sequential space and hence X is a sequential space.

Theorem 14. Let (X, τ) be a g-sequential space, (Y, σ) be a topological space and let f : X → Y be a map. Then f is strongly g-continuous if and only if f is sequentially g-continuous.

Definition 15. A map f :(X, τ) → (Y, σ) is said to be sequentially g-quotient if it satisfies the following: A is sequentially g-closed in Y if and only if f −1(A) is sequentially g-closed in X.

The following Theorem 16 shows that necessary and sufficient condition for a g-sequential space in terms of a sequentially g-quotient map

Theorem 16. Let (X, τ) be a topological space. Then X is g-sequential if and only if each quotient map on X is sequentially g-quotient.

In 2003, Mursaleen and Edely [11] introduced the statistical convergence by means of double sequences in a and the same was analyzed by Das and

Malik [4]. Moricz´ [10] introduced the notion of statistical convergence of multiple sequences in a metric space.

In the fourth chapter entitled, “Statistical convergence of double se- quences”, we give the definition of statistical limit points and statistical cluster points and investigate their properties in double sequences. Also, we discuss the re- lation between the statistical limit points and statistical cluster points. We introduce

6 A study on sequential spaces in topological and ideal topological spaces the s-convergent topology in double sequences. Also, we derive s-convergent topol- ogy determined by the family of all s-convergent sequences in s-sequential space.

Finally, we establish a property of s-sequential space.

Definition 17. Let A ⊂ N × N. Put A(m, n) = {(k, l) ∈ A | k ≤ m, l ≤ n}. Then |A(m, n)| |A(m, n)| we call δ(A) = lim inf and δ¯(A) = lim sup , the lower and m,n→∞ mn m,n→∞ mn upper asymptotic density of A, respectively. If δ(A) = δ¯(A), then d(A) = δ(A) = |A(m, n)| lim is called the double asymptotic density (or double natural density) m,n→∞ mn of A.

Definition 18. A double sequence (xmn) in a topological space (X, τ) is said to converge statistically (or shortly, s-converge) to x ∈ X, if for every neighborhood U of x, d({(m, n) ∈ × | xmn ∈/ U}) = 0. In this case, we write x = s- lim xmn N N m,n→∞ s or (xmn) −→ x.

Theorem 19. In a topological space (X, τ), the following statements hold for a s- convergent double sequence.

(a) For every x ∈ X, the double sequence (xmn) = x for all (m, n) ∈ N × N, s- converges to x.

(b) Addition of finite number of terms to a s-convergent double sequence affects neither its s-convergence nor s-limit to which it s-converges.

(c) If (xmn) is a s-convergent double sequence in A ∪ B which s-converges to x, where A and B are two non empty disjoint subsets in X, then there exists a double sequence (ymn) either in A or in B consisting of infinitely many terms of (xmn)

7 A study on sequential spaces in topological and ideal topological spaces s-converging to x.

(d) If (xmn) is a double sequence s-converging to a point x and (xp) be formed such that, for each p ∈ N, xp equals to some xmn where m > p and n > p, then the double sequence (ymn) where for each m ∈ N, ymn = (xm) for all n ∈ N, s-converges to x

Theorem 20. Let X be a given set and let φ be a class of double sequences over

X. Let the members of φ be called s-convergent double sequences and let each s- convergent double sequence be associated with an element of X called the s-limit of the s-convergent double sequence subject to the conditions (a) to (d) as stated in

Theorem 19. Now, let a subset A of X be called open if and only if no s-convergent double sequence lying in X \ A has any s-limit in A. Then the collection of open sets τ thus obtained forms a topology on X.

0 0 Theorem 21. Let (X, τs) be an s-sequential space and τs be the s-convergent topology

0 on X determined by the family of all s-convergent sequences in (X, τs). Then (X, τ) is also a s-sequential space.

In 1993, Hong [7] studied and investigated Fr´echet spaces and sequential con- vergence groups. Motivating this, we extend this concept to ideal topological spaces in chapter five entitled, “Convergence in ideal topological spaces”. First, we

introduce the definitions of sequential I-convergence structure (SIC), cLI and se-

quential I-convergence groups. Then we derive some properties of the operator cLI and sequential I-convergence groups. Finally, we discuss the concept of sequential

8 A study on sequential spaces in topological and ideal topological spaces

I-convergence spaces, I-Fr´echet-Urysohn spaces and study their properties. Se- quential I-convergence spaces and I-Fr´echet spaces are completely determined by

I-convergent sequences.

Definition 22. Let (X, τ) be a topological space with an ideal I on N and S[X] be the set of all sequences in X. A non empty subfamily LI of S[X] × X is called a sequential I-convergence structure (SIC) on X if it satisfies the following properties:

(SIC1) For each x ∈ X, ((x), x) ∈ LI where (x) is the constant sequence whose n-th term is x for all indices n ∈ N,

(SIC2) If ((xn), x) ∈ LI , then ((xni ), x) ∈ LI for each subsequence (xni ) of (xn).

(SIC3) Let x ∈ X and A ⊂ X. If ((xn), x) ∈/ LI for each (xn) ∈ S[A], then

((yn), x) ∈/ LI for each (yn) ∈ S[{y ∈ X|((xn), y) ∈ LI for some (xn) ∈ S[A]}].

A topological space (X, τ) together with a sequential I-convergence structure LI on X is called sequential I-convergence space and is denoted by (X,LI ). SCI[X] denote the set of all sequential I-convergence structures on X.

Definition 23. Let (X, τ) be a topological space with an ideal I on N. For each

LI ∈ SIC[X], define a map cLI of the P(X) of X into itself as follows:

cLI (A) = {x ∈ X | ((xn), x) ∈ LI for some (xn) ∈ S[A]}.

I Let L(cLI ) denote the set of all pairs ((xn), x) ∈ S[X] × X such that (xn) −→ x in

(X, cLI ).

The following Theorem 24 gives the properties of the operator cLI .

9 A study on sequential spaces in topological and ideal topological spaces

Theorem 24. Let (X, τ) be a topological space with an ideal I on N and A, B ⊂ X.

Then the following hold.

(a) cLI (∅) = ∅.

(b) A ⊂ cLI (A).

(c) A ⊂ B ⇒ cLI (A) ⊂ cLI (B).

(d) cLI (cLI (A)) = cLI (A).

(e) cLI (A ∪ B) = cLI (A) ∪ cLI (B).

Definition 25. Let (X,LI ) be a sequential I-convergence space. Then X is said to satisfy (∗))-property if for ((xn), x) ∈ LI and ((xnm), xn) ∈ LI for each n ∈ N, it is possible to choose a cross-sequence (xnm(n)) in the double sequence (xnm) such that (i) ((xnm(n)), x) ∈ LI (ii) m(n) ≥ n for all n ∈ N and (iii) ((xnk(n)), x) ∈ LI if k(n) ≥ m(n) for all n ∈ N.

The following Theorem 26 shows that the condition (∗))-property is sufficient for the product of two sequential I-convergence spaces to be a sequential I-convergence space.

Theorem 26. Let (X,LIX ) and (Y,LIY ) be any two sequential I-convergence spaces

satisfying (∗)-property and let LIX ×LIY = {((xn, yn), (x, y)) | ((xn), x) ∈ LIX and ((yn), y) ∈

LIY }. Then (X × Y,LIX × LIY ) is a sequential I-convergence space satisfying (∗)- property

Definition 27. A sequential I-convergence space (X,LI ) is called Hausdorff if LI satisfies the following property: If ((xn), x) ∈ LI and ((xn), y) ∈ LI , then x = y.

10 A study on sequential spaces in topological and ideal topological spaces

Definition 28. Let (X,LI ) be a Hausdorff sequential I-convergence space satisfying

(∗))-property and let · be a commutative group operator on X. The triple (X, ·,LI ) is called a sequential I-convergence group if it satisfies the following property:

−1 −1 (SIG) For each ((xn), x) ∈ LI and ((yn), y) ∈ LI , ((xnyn ), xy ) ∈ LI

Definition 29. Let (X, ·,LI ) be a sequential I-convergence group. A sequence

(xn) ∈ S[X] is called I-Cauchy if for each pair of subsequences (xni ) and (xnj ) of

(x ), ((x x−1), e) ∈ L where e is the identity element of the group (X, ·). n ni nj I

∗ ∗ Moreover, we will construct a sequential I-convergence structure LI on X . Let

∗ ∗ ∗ LI be the set of all pairs ((αn), α) ∈ S[X ] × X satisfying the condition that there

−1 −1 exists (xn) ∈ CI [X] such that αmxm = α[(xn)] for each m ∈ N.

∗ Theorem 30. Assume that LI satisfies the following condition:

∗ ∗ ∗ (∗∗) Let α ∈ X and (αnm) be a double sequence in X with ((αnm), α) ∈ LI for each n ∈ N. It is possible to choose (xn) ∈ CI [X] satisfying the property that for

−1 −1 each p ∈ N, there exists a sequence (xnp(m)) of (xn) such that αpmxnp(m) = α[(xn)]

∗ ∗ for all m ∈ N. Then (X , ∗,LI ) is an I-complete sequential I-convergence group containing (X, ·,LI ).

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