EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 16, No. 1, 2023, 1-4 ISSN 1307-5543 – ejpam.com Published by New York Business Global More on Ideal Rothberger Spaces Asli Guldurdek College of Engineering and Technology, American University of the Middle East, Egaila, 54200, Kuwait Abstract. The aim of this note is to provide an answer to a question posted in a recent paper. In 2018, after introducing the notion of Ideal Rothberger space, author examines some properties of these spaces. Also there has been a comparison of the spaces (X, τ), and (X, τ∗) in terms of being (ideal)Rothberger. According to this, it is shown that if (X, τ∗) is a Rothberger space, then (X, τ) is also Rothberger. Therefore, naturally it is asked that, if one can find some extra conditions for ideal I, then the opposite also holds. Thus, for which ideal I, an I-Rothberger space (X, τ) implies an I-Rothberger space (X, τ∗)? In this work it has been proved that I is a σ-ideal, and τ is compatible with I, which provides the solution. 2020 Mathematics Subject Classifications: 54A05, 54D20 Key Words and Phrases: Ideal Topological Space, Ideal Rothberger Space 1. Introduction and Preliminaries Extending topological spaces by adding ideals, has been done since Vaidyanathaswamy [7], and Kuratowski [3]. An ideal I on a set X is defined to be a nonempty collection of subsets of X, which is closed under the subset and finite union operations. A topological space (X, τ) with an ideal I defined on X is donoted by (X, τ, I). By using an ideal on a topological space, a local function is defined in [3] as follows: Definition 1. [3] Let (X, τ) be a topological space, and let I be an ideal on X. Then the local function A∗(I, τ) of A ⊂ X is defined as: A∗(I, τ) = {x ∈ X | A ∩ U /∈ I for every U ∈ τ(x)}, where τ(x) = {U ∈ τ | x ∈ U}. Using the notation of the previous definition, it can easily be seen that, by saying, for every A ⊂ X, c∗(A) = A ∪ A∗(I, τ), one defines a Kuratowski closure operator c∗.This closure operator induces a topology on X (see [2]). Definition 2. For an ideal I of subsets of a topological space (X, τ), the topology on X induced by the closure operator c∗ is denoted by τ∗(I, τ) or for simplicity, τ∗ if this does not lead to misunderstanding. DOI: https://doi.org/10.29020/nybg.ejpam.v16i1.4625 Email address: asli.guldurdek@aum.edu.kw (A. Guldurdek) https://www.ejpam.com 1 © 2023 EJPAM All rights reserved. A. Guldurdek / Eur. J. Pure Appl. Math, 16 (1) (2023), 1-4 2 On the other hand, examining the covering properties of topological spaces is also an attractive area for topologists. One of these covering properties, which is introduced in [5], [6], is being a Rothberger Space. Definition 3. [5], [6] A space X is Rothberger if for every sequence {Un | n ∈ N} of open covers of X there exists a sequence {Un | n ∈ N} such that Un ∈ Un for every n ∈ N, and X = ⋃ n∈N Un. So, in [1] the notions of ideal topological space and Rothberger space were brought together, and resulted in Ideal Rothberger(I-Rothberger) Spaces. Definition 4. [1] Let (X, τ, I) be an ideal topological space. (X, τ) is said to be I- Rothberger or Rothberger with respect to I, if for every sequence {Un | n ∈ N} of open covers of X there exists a sequence {Un | n ∈ N} such that Un ∈ Un for every n ∈ N, and X \ ⋃ n∈N Un ∈ I. In [1] besides exploring properties, and weak forms of I-Rothberger spaces, a theorem about the transition between (X, τ), and (X, τ∗) is given as following: Theorem 1. Let (X, τ) be a topological space, I be an ideal on X, and (X, τ∗) be a Rothberger space. Then (X, τ) is also Rothberger, and since every Rothberger space is shown to be I-Rothberger, therefore (X, τ) is indeed an I-Rothberger space. Note that, the proof is clear via the fact that τ ⊂ τ∗. Following the previous theorem, an example which shows that being I-Rothberger for (X, τ) does not imply being Rothberger for (X, τ∗). Example 1. [1] Let X be the set of real numbers R, the topology τ be the usual topology of R, and the ideal I be the power set P(R). It is clear that, (R, τ) is P(R)-Rothberger. On the other hand τ∗ on R is the discrete topology, which does not qualify as Lindelöf, therefore, it is not Rothberger. Based on these, a question is posted in [1]. [1] What extra conditions the ideal I might have, in order to provide the converse of the previous theorem? In this paper, the condition for the ideal I is provided. 2. Main Result Given a topology τ on a set X and an ideal I of subsets of X, such that I is compatible with τ , τ∗ denotes the topology on X consisting of all sets of the form U \A where U ∈ τ and A ∈ I. In 2018, the notion of an I-Rothberger space was introduced. It is known that if (X, τ∗) is a Rothberger space, so is (X, τ). However, a satisfactory answer to the A. Guldurdek / Eur. J. Pure Appl. Math, 16 (1) (2023), 1-4 3 question under which conditions on I the reverse implication also holds is still unknown. The main theorem of the article asserts that it holds if I is a σ-ideal of subsets of X such that I is compatible with a topology τ on X, and the space (X, τ) is I-Rothberger, so is (X, τ∗). This gives a partial answer to the above-mentioned question. Before providing this condition, some basic definitions, and theorems to be used are included: Definition 5. [2] An ideal I is said to be a σ-ideal if it is countably additive, that is, if In ∈ I, for each n ∈ N, then ⋃ {In|n ∈ N} ∈ I. Definition 6. [4] Let (X, τ) be a topological space with an ideal I. The topology τ is indicated as compatible with the ideal I, which is denoted by τ ∼ I, if the following holds for every A ⊂ X: if for every x ∈ A, there exists a set U ∈ τ(x) such that U ∩A ∈ I, then A ∈ I. Theorem 2. [4] Let (X, τ) be a topological space, I be an ideal on X, and τ be compatible with I. A set is closed in τ∗ if and only if it is the union of a set which is closed with respect to τ and a set from the ideal I. Corollary 1. [2] Let (X, τ) be a topological space and I be an ideal on X, with τ ∼ I. Based on the previous theorem it is known that every closed set K can be written as K = F ∪ I, where F is closed with respect to τ , and I ∈ I. Then every τ∗-open set U will be the complement of the sets of this type and hence will have the form: U = G \ I, where G ∈ τ , and I ∈ I. Finally, the answer to the question that was posted: Theorem 3. Let (X, τ) be a topological space, I be a σ-ideal on X, and τ ∼ I. If (X, τ) is I-Rothberger, then (X, τ∗) is also an I-Rothberger space. Proof. Let {Un | n ∈ N} be a sequence of open covers of X with respect to topology τ∗. So; Un = {Uα n | α ∈ ∆n} ; for every n ∈ N, and Uα n ∈ τ∗. By compatibility: Uα n = Gα n \ Iαn , where Gα n ∈ τ , and Iαn ∈ I. Clearly Gn = {Gα n | α ∈ ∆n} is an open cover of X, with respect to topology τ , and so {Gn | n ∈ N} is a sequence of open covers. Since (X, τ) is I-Rothberger, there exists a sequence {αn | n ∈ N} such that, for every n ∈ N, αn ∈ ∆n, and J = X \ ∞⋃ n=1 Gαn n ∈ I. On the other hand, since I is a σ-ideal, it is accepted that K = ∞⋃ n=1 Iαn n ∈ I. So; J ∪K = (X \ ∞⋃ n=1 Gαn n ) ∪ ( ∞⋃ n=1 Iαn n ) ∈ I. And clearly; REFERENCES 4 X \ ∞⋃ n=1 Uαn n ⊆ (X \ ∞⋃ n=1 Gαn n ) ∪ ( ∞⋃ n=1 Iαn n ). Finally; X \ ∞⋃ n=1 Uαn n ∈ I, so (X, τ∗) is an I-Rothberger space. 3. Conclusions In this work, a question posted in [1] is partially answered. So, one may still search for a weaker condition for a possible future work. Acknowledgements The author would like to thank the anonymous referee for their useful comments. References [1] A. Güldürdek. Ideal Rothberger spaces. Hacettepe Journal of Mathematics and Statis- tics, 47:69–75, 2018. [2] D. Janković and T.R. Hamlett. New topologies from old via ideals. The American Mathematical Monthly, 97(4):295–310, 1990. [3] K. Kuratowski. Topology. Academic Press, New York, NY, 1966. [4] O. Njastad. Remarks on topologies defined by local properties. Avh. Norske Vid.-Akad. Oslo I (N.S.), 8:1–16, 1966. [5] F. Rothberger. Eine verschärfung der eigenschalft. Fund. Math., 3:50–55, 1938. [6] M. Scheepers. Combinatorics of open covers(I): Ramsey theory. Topology Appl., 69:31– 62, 1996. [7] R. Vaidyanathaswamy. The localisation theory in the set topology. Proc.Indian Acad. Sci., 20:51–61, 1945.