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This paper is devoted to studies of I wQN-spaces and some of their cardinal characteristics. Recently, Šupina in [32] proved that I is not a weak P-ideal if and only if any topological space is an I QN-space. Moreover, under p = c he constructed a maximal ideal I (which is not a weak P-ideal) for which the notions of I QN-space and QN-space do not coincide. In this paper we show that, consistently, there is an ideal I (which is not a weak P-ideal) for which the notions of I wQN-space and wQN-space do not coincide. This is a partial solution to [6, Problem 3.7] . We also prove that for this ideal the ideal version of Scheepers Conjecture does not hold (this is the first known example of such weak P-ideal). We obtain a strictly combinatorial characterization of non ( I wQN-space ) similar to the one given in [32] by Šupina in the case of non ( I QN-space ) . We calculate non ( I QN-space ) and non ( I wQN-space ) for some weak P-ideals. Namely, we show that b ≤ non ( I QN-space ) ≤ non ( I wQN-space ) ≤ d for every weak P-ideal I and that non ( I QN-space ) = non ( I wQN-space ) = b for every F σ ideal I as well as for every analytic P-ideal I generated by an unbounded submeasure (this establishes some new bounds for b ( I , I , Fin ) introduced in [31] ). As a consequence, we obtain some bounds for add ( I QN-space ) . In particular, we get add ( I QN-space ) = b for analytic P-ideals I generated by unbounded submeasures. By a result of Bukovský, Das and Šupina from [6] it is known that in the case of tall ideals I the notions of I QN-space ( I wQN-space) and QN-space (wQN-space) cannot be distinguished. Answering [6, Problem 3.2] , we prove that if I is a tall ideal and X is a topological space of cardinality less than co v ⁎ ( I ) , then X is an I wQN-space if and only if it is a wQN-space. [ABSTRACT FROM AUTHOR]

Abstract In this paper we primarily introduce certain new versions of groupability, namely star- I -groupability which is introduced using both star operation and the notion of ideals and then a more general notion of R -groupability. We establish several results regarding these groupability notions in the process of extending certain existing results on groupability. [ABSTRACT FROM AUTHOR]

We consider four classes of topological spaces which are defined with the aid of convergence with respect to ideals on N . All these classes are subclasses of countably compact spaces, and two of them are also subclasses of sequentially compact spaces. In the first part of the paper (Sections 1 and 2 ) we prove some properties of these classes. In the second part of the paper (Sections 3 and 4 ) we focus on spaces defined by two particular ideals connected with well known theorems in combinatorics, namely van der Waerden's theorem and Hindman's theorem. The main aim of this part of the paper is to show that two classes of the considered spaces coincide for the van der Waerden ideal and Hindman ideal respectively. [ABSTRACT FROM AUTHOR]

In this paper we primarily consider the ideal analogue of Hurewicz property introduced recently by Das in [6] and subsequently introduce the ideal analogues of several versions of this very important property studied in the last decade. We establish certain inter relationships between the new notions presenting suitable examples as far as possible as well as prove many related results and in particular preservation properties. [ABSTRACT FROM AUTHOR]

In this paper we discuss the following Tkachuk's question in the sense of ideal convergence [16,18] : Is any Tychonoff connected sequential space a quotient image of a connected metric space? It is proved that let I be an ideal on the set N then a topological space X is an I -connected space with an I - csf -network if and only if X is a continuous I -covering image of a connected metric space. It follows that a topological space X is a connected I -sequential space with an I - csf -network if and only if X is a quotient I -covering image of a connected metric space. [ABSTRACT FROM AUTHOR]

In this paper we follow the line of recent works of Das [4,5,7] (see also [3] for similar investigation), where a more general approach was made to study certain results on open covers and selection principles by using the notion of ideals and ideal convergence, which automatically extends similar classical results (where finite sets are used), and also recent statistical variants studied by Di Maio and Kočinac [10] . Here we further introduce the notions of cω -covers and I -large covers which extend the notions of ω -covers and large covers in a topological space. We then study the I -groupability of different open covers and some of its consequences. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we primarily consider the notion of -covers recently introduced in [11] as well as related notion of -covers [9] and make some further investigations. In particular we consider the notion of κ-covers studied by Di Maio, Kočinac, Meccariello and Caserta in [8,18], and establish some results involving -covers ( -covers) and κ-covers. Our results extend the earlier results proved in [6,8,18,29] and present a more general version using the notion of ideals. [Copyright &y& Elsevier]

Let L be a completely regular frame and, as customary, let R L denote the ring of continuous real-valued functions on L. In the first part of the paper we characterize the maximal regular ideal of R L. We show that it consists precisely of the functions α such that the open sublocale of L associated with coz α is clopen and is a P -frame. We also give a characterization of this ideal in terms of the notion of the localic remainder. In contrast, the maximal regular ideal of the ring Z L of continuous integer-valued functions on L is shown to be always the zero ideal. Purity and regularity of ideals are germane in the description and characterization of the maximal regular ideal of R L. It is for this reason that in the last part of the paper we consider ideals of ideals of R L because, as we prove, it is precisely the pure ideals of R L all of whose ideals are ideals in the whole ring. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we make some further investigations on ideal convergence and in particular we concentrate on -limit points and -cluster points. We try to establish the characterization of the set of -limit points (which has not been done in any structure so far) and show that this set can be characterized as an -set for a large class of ideals, namely analytic P-ideals and then make certain interesting observations on -cluster points. [Copyright &y& Elsevier]

Abstract: In this paper we continue our investigation of nets using ideals in line of our earlier work where we had studied -Cauchy nets and asked when -Cauchy nets in complete uniform spaces are -convergent in line of a problem by Di Maio and Kočinac who asked in 2008 when statistically Cauchy sequences are statistically convergent in uniform spaces. To answer this, here we consider another type of Cauchy condition of nets, namely -Cauchy condition and examine its basic properties and in particular its relation with the concept of -Cauchy nets. This helps us to give an answer to the above mentioned open question. [Copyright &y& Elsevier]

Abstract: In this paper we extend the idea of usual Cauchy condition of nets to I-Cauchy condition by using the concept of ideals. This Cauchy condition arises naturally from the notion of I-convergence of nets introduced by Lahiri and Das (2008). As the underlying structure for the whole study we take a uniform space so that our notion and results extend the idea of statistical Cauchy sequences very recently introduced in uniform spaces by Di Maio and Kočinac (2008). In particular we try to give partial answers to an open problem posed by Di Maio and Kočinac and examine the relationship between this new Cauchy condition and usual completeness of a uniform space. [Copyright &y& Elsevier]

Abstract: Given an infinite group G and an infinite cardinal , we say that a subset A of G is κ-large (κ-small) if there exists such that ( is κ-large for each ). The subject of the paper is the family of all κ-small subsets. We describe the left ideal of the right topological semigroup βG determined by . We study interrelations between κ-small and other (-small and κ-thin) subsets of groups, and prove that G can be generated by some 2-thin subsets. We partition G in countable many subsets which are κ-small for each . We show that is dual to provided that either κ is regular and , or G is Abelian and κ is a limit cardinal, or G is a divisible Abelian group. [Copyright &y& Elsevier]

We denote by C s (X) the set C (X) of real-valued continuous functions defined on X endowed with the topology of the uniform convergence on the closed separable subspaces of X. In this paper we continue the study of C s (X) initiated in Pseudouniform topologies on C (X) given by ideals (Pichardo-Mendoza et al. (2013) [6]). We prove that C s (X) is a k -space if and only if C s (X) is metrizable, and that compactness, sequential compactness and countable compactness coincide in subspaces of C s (X). In addition, we study the cellularity, density, weight and character of C s (X). We prove that (1) d ((R 2 λ ) s) = λ if λ = λ ω , and (2) the Continuum Hypothesis is equivalent to the statement: Every non-separable space X satisfies χ (C s (X)) = w (C s (X)). [ABSTRACT FROM AUTHOR]

Let X be a Tychonoff space. Associated with every subset A ⊆ β X is the ideal O A of the ring C (X) consisting of all functions that vanish in a neighborhood of X ∩ A. Now, viewing Tychonoff spaces as objects in the category CRLoc of completely regular locales, we have ideals of the form O A , where A is a sublocale of βX. In this paper we study properties of such ideals not only for Tychonoff spaces, but for any object in CRLoc. Carrying out the discussion in this category, we have more function rings (they are denoted R L , for any L ∈ CRLoc) than the class of the rings C (X). Pure ideals of C (X) are known to be exactly the ideals O A , for A a closed subset of βX. We characterize the spaces for which the ideals O A , for A a closed subset of X (note that it need not be closed in βX) are pure ideals. They properly contain the normal ones. We describe the socle of any ring R L as the ideal O A , with A equal to the join of all nowhere dense sublocales of βL. We show that the socle is zero precisely when βL is dense in itself, and essential if and only if the smallest dense sublocale of βL has a complement in the lattice of sublocales of βL. If βL is scattered, then this is so if and only if L has a smallest nowhere dense sublocale. [ABSTRACT FROM AUTHOR]