Research Article | Open Access
A. A. Azzam, "A New Closed Set in Topological Spaces", Mathematical Problems in Engineering, vol. 2021, Article ID 6617224, 4 pages, 2021. https://doi.org/10.1155/2021/6617224
A New Closed Set in Topological Spaces
Our purpose of this work is to implement a class of -closed sets, which is property placed among the classes of semiclosed sets and -closed sets. The relations with other concepts directly or indirectly joined with generalized closed sets are inspected. In addition, as an application, using the notion of -closed sets, we give a brief expansion of a new space named -space.
1. Introduction and Preliminaries
Topological space has become one of the most important spaces that help to solve many of the contemporary problems. It is the reference for measuring the descriptive rather than quantitative. Also, closed sets are essential matters in a space that carries topology. For instance, one can know the topology on a set by using either the axioms for the closed sets or the Kuratowski closure axioms. In 1970, Levine  initiated the study of so-called generalized closed sets. By definition, a subset of the space which carries topology is named generalized closed set if the closure of any subset of is included in each open superset of . This notion has been studied extensively in recent years by many topologists because generalized closed sets are not only natural generalizations of closed sets. Furthermore, the study of generalized closed sets also extends new depictions of some known classes of spaces, for instance, the class of extremely disconnected spaces. In 1987, Bhattacharyya and Lahiri  introduced a new class of sets named semigeneralized closed sets using semiopen sets of Levine  which obtained various properties corresponding to . In 1990, Arya and Nour  defined the generalized semiclosed sets. Dontchev  has introduced -closed sets by generalizing semi-preopen sets. Generalized closed sets in bitopological spaces have been introduced by Fukutake . More recently, Kumar  introduced -closed sets in topological spaces. John and Sundaram  introduced and studied the concept of -closed sets and -space concerning bitopological space. Noiri and Rajesh  introduced generalized closed sets with respect to an ideal in bitopological spaces. Noiri and popa  studied the relation between -closed sets and --closed sets in ideal topological spaces. In 2018, Al-Saadi  discussed the concept of strongly -closed sets and strongly -spaces in bitopological spaces. In the present work, we introduce the notion of -closed sets and find some basic properties for it. We are also showing that this class lies among the class of semiclosed sets and the class of - closed sets. Applying the sets, we introduce a new space called -space. Let us review some of the standard facts on near closed sets as the next.
Definition 1. A subset of a space which carries topology is named as follows:(i)Semiclosed  if ,(ii)Semiopen  if is semiclosed or equivalently, if ,(iii)Semi-preclosed  if ,(iv)Semi-preopen  if is semi-preclosed or equivalently, if .(v)Generalized closed  if whenever and is open set in . The semiclosure [12, 14] of a subset of , denoted by , briefly , is defined to the intersection of all semiclosed sets containing . The semi-interior of , denoted by , is defined by the union of all semiopen sets contained in . A number of definitions and depictions have been handled in [1, 2, 4, 5, 7] with respect to generalized closed sets or -space.
2. Basic Properties of -Closed Sets
In section 2, we give a brief exposition of -closed sets, some of their properties, and relations with other known classes of subsets.
Definition 2. Let be a nonempty set that carries topology ; a subset of is called -closed if whenever and is a -open set in .
Theorem 1. Let be a nonempty set that carries topology . Then, the next declaration is verified.(i)Each semiclosed set is -closed in (ii)Each -closed set is -closed in (iii)Each -closed set is closed and hence closed in (iv)Each -closed set is -closed in
Proof. (i)Let be a semiclosed set and be any -open containing . Since is semiclosed, for each subset of . Therefore, , and hence is -closed.(ii)It is obvious from inclusion for each subset of .(iii)It is clear that for each subset of , and each open set is -open.(iv)From for each subset of , the proving is clear.
Remark 1. Figure 1 shows the connections results among -closed and different types of other sets.
In Remark 1, the relationships cannot be reversible as the next instance shown.
Example 1. Let carry topology .(i)If , then is -closed but neither semiclosed nor -closed.(ii)If , then is closed and closed but not -closed.(iii)If , then is -closed but not -closed. Also, is -closed but not semiclosed.
Remark 2. The following examples from (i) to (iii) show that the concept of -closedness is independent from -closedness, closedness, -closedness, and -closedness.(i)Let and . In a space , a subset is -closed but it is not -closed. Also, a subset is -closed but neither closed nor -closed in .(ii)Let be the same space in Example 1(ii). A subset is semiclosed and -closed but it is not -closed.(iii)Let be the usual topology on the real line . One can deduce that the open interval is closed but not -closed.
Theorem 2. Let and be two -closed subsets of . Then,(i)The union of two -closed sets is also -closed in (ii)If is -open and -closed, then is semiclosed in (iii)If is -closed in , then does not contain any nonempty -closed set(iv)If is -closed of such that , then is also -closed of (v)For each is -closed or is -closed of (vi)Every subset is -closed of if and only if every -open set is semiclosed
Proof. (i)Let and be two -closed sets and be any -open set such that . Then, .(ii)Since is -open and -closed, . Hence, is semiclosed.(iii)Assume that there exists a -closed set such that . Then, and so . Hence, we have that and because .(iv)Let be a -open set such that . Then, we have that and . Therefore, is -closed in .(v)If is not -closed, then is not -open. Therefore, is -closed in .(vi)Let be a -open set. Then, we have that and hence is semiclosed. Conversely, let be a subset and a -open set such that . Then, , and hence is -closed.
Theorem 3. A subset of space is -closed if and only if contains no nonempty -closed set in .
Proof. Let be a nonempty -closed set of such that . Since is -open and is -closed, , i.e., . Then, and hence , which is a contradiction.
Conversely, let be a -open set such that . Suppose that is not contained in . Then, by Remark 1, is nonempty -closed and , which is a contradiction.
Remark 3. The intersection of two -closed sets need not be -closed as the following instance shows.
Example 2. Let with a topology . Put and . Then, and are -closed but is not -closed in .
3. -Open Sets
In the third section, we explain the definition of -open and related by -closed with some examples.
Definition 3. A subset of a space which carries topology is named -open if is -closed.
Remark 4. For A subset of a space which carries topology , .
Theorem 4. A subset is -open in if and only if , whenever and is -closed in .
Proof. Let be a -closed of and . Then, is -open and . Since is -closed, , that is, . Hence, .
Conversely, let be a -open set of and . Since is a -closed set contained in , by hypothesis , . By Remark 4, . Hence, is -closed, and so is -open.
Theorem 5. A subset is -closed in if and only if is -open.
Proof. Let be a -closed set of such that . Since is -closed, by Theorem 4, that and is -open.
Reciprocally, let , we have as -open in . Now, and since is -closed and is -open, . Then, or , and hence is -closed.
Example 3. Let with topology . In , the set is -open.
As an application of the concept of -closed sets, we introduce the following space called -space.
Definition 4. A space which carries topology is called a -space if each -closed set is closed in .
Remark 5. The spaces and are independent as seen from the next instances.
Example 4. Let with topology . Then, the space is -space but not a -space.
Example 5. Let with . Then, is -space but not -space.
Popularizations of closed sets in point-set topology will give some new topological properties (for instance, separation axioms, compactness, connectedness, and continuity) and a brief expansion of a new space named -space which have been found to be very beneficial in the study of certain objects of digital topology. Thus, we may stress once more the importance of -closed sets as a branch of them and the possible application in computer [15–17] and quantum.
The data are included only in references of manuscript.
Conflicts of Interest
The author declares that there are no conflicts of interest.
- N. Levine, “Generalized closed sets in topology,” Rendiconti del Circolo Matematico di Palermo, vol. 19, pp. 89–96, 1970.
- P. Bhattacharyya and B. K. Lahiri, “Semi-generalized closed sets in toplogy,” Indian Journal of Mathematics, vol. 29, pp. 375–382, 1987.
- N. Levine, “Semi-open sets and semi-continuity in topological spaces,” The American Mathematical Monthly, vol. 70, no. 1, pp. 36–41, 1963.
- S. P. Arya and T. Nour, “Charaterizations of s-normal space,” Indian Journal of Pure and Applied Mathematics, vol. 21, pp. 717–719, 1990.
- J. Dontchev, “On generaizing semi-preopen sets,” Mem. Fac. Sci. Kochi Univ. Ser.A Math, vol. 16, pp. 35–48, 1995.
- T. Fukutake, “On generalized closed sets in bitopological spaces,” Bulletin of Fukuoka University of Education, vol. 35, pp. 19–28, 1985.
- V. Kumar, “g-closed sets in topological spaces,” The Allahabad Mathematical Society, vol. 18, pp. 99–122, 2003.
- M. S. John and P. Sundaram, “-closed sets in bitopological spaces,” Indian Journal of Pure and Applied Mathematics, vol. 35, no. 1, pp. 71–80, 2004.
- T. Noiri and N. Rajesh, “Generalized closed setswith respect to an ideal bitopological spaces,” Acta Mathematica Hungarica, vol. 125, 2009.
- T. Noiri and V. popa, “Between -closed sets and --closed sets in ideal topological spaces,” Rendiconti del Circolo Matematico di Palermo, vol. 59, no. 2, pp. 251–260, 2010.
- H. Al-Saadi, “Strongly -closed sets and strongly spaces in bitopological spaces,” Hacettepe Journal of Mathematics and Statistics, vol. 47, no. 1, pp. 37–45, 2018.
- S. G. Crossely and S. K. Hildebrand, “Semi-closure,” Texas Journal of Science, vol. 22, pp. 99–112, 1971.
- D. Andrijevic, “Semi-preopen sets,” Matematički Vesnik, vol. 38, no. 1, pp. 24–32, 1986.
- N. Biswas, “On characterizations of semi-continuous functions,” Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali, vol. 8, pp. 399–402, 1970.
- M. S. El Naschie, “On the uncertainty of Cantorian geometry and two, slit Experiment, Chaos,” Chaos, Solitons & Fractals, vol. 9, no. 3, pp. 517–529, 1998.
- E. D. Kalimsky, R. Kopperman, and P. R. Meyer, “Computer graphics and connected topologies on finite ordered sets,” Topology and its Applications, vol. 36, pp. 1–17, 1990.
- T. Y. Kony, R. Kopperman, and P. R. Meyer, “A topological approach to digital toplogy,” The American Mathematical Monthly, vol. 98, pp. 901–917, 1991.
Copyright © 2021 A. A. Azzam. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.