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A Study on Sequential Spaces in Topological and Ideal Topological Spaces

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A Study on Sequential Spaces in Topological and Ideal Topological Spaces 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 Mathematics 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 sequences, se- quentially I-convergence structure and generalized convergence of a sequence 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 limit 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 metric 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 topological space. A function [·]s−seq of the power s set P(X) to itself defined by for each subset A of X,[A]s−seq = fx 2 X j (xn) −! x in (X; τ) for some sequence (xn) of points in Ag is called the s-sequential closure 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 = fX n [A]s−seq j A ⊂ Xg is a topology 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 = fX n [A]s−seq j A ⊂ Xg; then (X; τs−seq) is a s-Fr´echet-Urysohnspace. Moreover, for s each sequence (xn) of points in X and each p 2 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 [ f1g with 1 2= X and τ ∗ = τ [ fU ⊂ X∗ j 1 2 U; X − U is a closed, s-sequentially compact subset of (X; τ) g 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 2 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-sequential space. 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 = fx 2 X j x = glim xn and (xn) 2 S[A] \ cg(X)g 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 metric space 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.
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