In this note we show by producing counter examples that some results which appeared in the articles by Jankovic and Hamlett [3] are incorrect

Esmaeel, Rana Bahjat

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                            ISSN 2347-192 1514 | P a g e                                                          A p r i l  2 5 ,  2 0 1 4                                                            Short Communication A note on "The Ideal Generated by Codense Sets and the Banach Localization Property" M. E. Abd El-Monsef, A. A. Nasef and R. B. Esmaeel  Abstract In this note we show by producing counter examples that some results which appeared in the articles by Jankovicʹ and Hamlett [3] are incorrect. Mathematics Subject Classification 2000: 54A05, 54A10. Keywords: Codense sets and boundary sets.                             Council for Innovative Research Peer Review Research Publishing System Journal: Journal of Advances in Mathematics Vol 8, No 2 editor@cirjam.org www.jam.cirworld.com, www.cirworld.com                              ISSN 2347-192 1515 | P a g e                                                          A p r i l  2 5 ,  2 0 1 4                                                            1. Introduction The notion of ideal topological spaces was studied by Kuratowski [5] and Vaidyanathaswamy [8]. Applications to various fields were further investigated by Jankovicʹ and Hamlett [4]; Dontchev et al. [2]; Mukherjee et al. [6]; Arenas et al.[1]; Nasef and Mahmoud [7], etc. In this note, we show that there are Mathematical errors of the claimed results of the paper of Jankovicʹ and Hamlett [3]. 2. Preliminaries and Some Examples Throughout the present note, (X, τ) will denote a topological space with no separation properties assumed. For a subset A of a topological space (X, τ), Cl(A), Int(A) and Bd(A) will denote the closure, the interior and the boundaries of A in (X, τ), respectively. An ideal I on a topological space (X, τ) is a nonempty collection of subsets of X which satisfies; (i) A∈ I and B A implies B∈ I, (hereditry), (ii) A∈ I and B∈ I implies A∪B∈ I, (finite additivity). It is interesting to note that if I is a proper ideal (i.e. X∉ I), then the collection of the complements of the members of I form a filter on X. This is why sometimes ideals are called dual filters. The following collections form important ideals [3, 4, 8] on a topological space (X, τ): 1. {Ø} or I{ø}: the trivial ideal. 2. Ƥ(X): the improper ideal. 3. Ƒ: (If) the ideal of all finite sets. 4. Ϲ: (Ic) the ideal of all countable subsets of X. 5. ϹD: (Icd) the ideal of all closed and discrete sets in (X, τ). i.e. Icd= {A X: Ad= Ø} where Ad is the derived set of A. 6. Ɲ: (In) the ideal of all nowhere dense sets. i.e. In=  {A X: Int(Cl(A))= Ø}. 7. Ϻ: (Im) the ideal of all first category (= meager) sets. (A∈Im iff A is a countable union of nowhere dense sets) i.e. Im= {A X: A= ∪{Bi , i∈ ; Int(Cl(Bi))= Ø}. 8. S: (Is) the ideal of all scattered sets (here X must be T0) or τD-space. 9. ʆ: (IL0) the ideal of all Lebesgue null sets (here X stands for the real line). 10. PB: the ideal of all parabounded subsets of (X, τ). 11. CDF: the ideal of all closed df- sets, where a subset A of a topological space (X, τ) is called discretely finite (= df- set) if for each x∈ A, there exists an open set U containing x such that U∩ A is finite. 12. IK: the ideal of relatively compact sets in (X, τ). (A is relatively compact iff Cl(A) is compact). 13. >A<: I(A) the principal ideal generated by any subset A of (X, τ).   i. e. I(A)= Ƥ(A)= {B X: B A}. Observation 2.1.                             ISSN 2347-192 1516 | P a g e                                                          A p r i l  2 5 ,  2 0 1 4                                                            In [3] D. Jankovicʹ and T.R. Hamlett showed that the collection Ϲ= {A X: Int(A)= Ø } of all codense subsets in a topological space (X, τ) formed an ideal on X and denoted by I(Ϲ). The following example shows that, in general the family of all codense subsets in a topological space (X, τ) need not be an ideal on X. Example 2.1. Let X= {a, b, c} with a topology τ ={X, Ø, {b,c}}. Then I(Ϲ)= {A X: Int(A)= Ø }= { Ø, {a}, {b}, {c}, {a, b}, {a, c}} which is not an ideal, since {b}, {c}∈ I(Ϲ) while {b, c} does not belongs to I(Ϲ). Observation 2.2. Jankovicʹ and Hamlett [3] showed that the collection Ɓ= {Bd(A): A X} of all boundaries of subsets of the space (X, τ) form an ideal and denoted by I(Ɓ). The following example shows that, the family of all boundaries of subsets of the space (X, τ) is not an ideal on X in general. Example 2.2. Let X be as in Example 2.1. with a topology τ= {X, Ø, {a}, {a, b}}. Then one can deduce that I(Ɓ)= { Ø, {c}, {b, c}} which is not an ideal, since {b} {b, c}∈ I(Ɓ) while {b} does not belongs to I(Ɓ). Observation 2.3. Jankovicʹ and Hamlett [4] claimed that I∪{X} is a topology on X, where I is an ideal on X. The following example shows that, I∪{X} is not a topology on X in general. Example 2.3. Let X=  be the set of all real numbers with an ideal I= {A  : A is finite subset of the natural numbers }. It is clear that {n}∈ I for each n∈ , but ∪{{n}: n∈ }=  ∉ I∪{X}. Hence, I∪{X} is not a topology on X. This note, I ∪{X} is a topology on X, is hold where X is finite set. 3. Conclusions The aim of this note is to point out that one assertion (The collections of all codense sets and the collections of all boundary sets in any topological space) in a previous paper by Jankovicʹ and Hamlett [D. Jankovicʹ and T. R. Hamlett, The ideal generated by codense sets and the Banach localization property, Coll. Math. Soc. Janos Bolyai, 55, Topology, Pecs, Hungary, (1989), 349- 358] not represent ideal by a counterexamples. And so we show that, the note I∪{X} is not a topology on X in general which was introduced by Jankovicʹ and Hamlett [D. Jankovicʹ and T. R. Hamlett, New Topologies from old via ideals. Amer. Math. Monthly 97(1990), 295- 310].  References [1] F. G. Arenas, J. Dontchev and M. L. Puertas, Idealization of some weak separation axioms. Acta Math. Hungar. 89(1- 2)(2000),47- 53. [2] J. Dontchev, M. Ganstar and D. Rose, Ideal resolvability. Topology and its Appl. 93(1949), 1- 16. [3] D. Jankovicʹ and T. R. Hamlett, The ideal generated by codense sets and the Banach localization property, Coll. Math. Soc. Janos Bolyai, 55, Topology, Pecs, Hungary, (1989), 349- 358. [4] D. Jankovicʹ and T. R. Hamlett, New Topologies from old via ideals. Amer. Math. Monthly 97(1990), 295- 310. [5] K. Kuratowski, Topology, Vol. I. New York: Academic Press, 1966. [6] M. N. Mukherjee, R. Bishwambhar and R. Sen, On extension of topological spaces in terms of ideals. Topology and its Appl. 154(2007), 3167- 3172. [7] A. A. Nasef and R. A. Mahmoud, Some applications via fuzzy ideals. Chaos. Solitons and Fractals, 13(2002), 825- 831. [8] R. Vaidyanathaswamy, The localisation theory in set topology. Proc. Indian Acad. Sci. 20(1945), 51-61.                              ISSN 2347-192 1517 | P a g e                                                          A p r i l  2 5 ,  2 0 1 4                                                             M. E. Abd El-Monsef, Department of Mathematics, Faculty of Seience, Tanta       University, Tanta, Egypt.        me_monsef@yahoo.com   A. A. Nasef, Department of Physics and Engineering Mathematics, Faculty of Engineering, Kafr El-Sheikh University, Kafr El-Sheikh, Egypt. nasefa50@yahoo.com  R. B. Esmaeel, Department of Mathematics, Ibn Al-Haitham College of Education, Baghdad University, Iraq. ranamumosa@yahoo.com 

Short Communication A note on "The Ideal Generated by Codense Sets and the Banach Localization Property"

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Short Communication A note on "The Ideal Generated by Codense Sets and the Banach Localization Property"

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