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For a Tychonoff space X, the lattice UX was introduced in L. A. Pérez-Morales, G. Delgadillo-Piñón, and R. Pichardo-Mendoza, The lattice of uniform topologies on C(X), Questions and Answers in General Topology 39 (2021), 65-71. In the present paper we continue the study of UX. To be specific, the present paper deals, in its first half, with structural and categorical properties of UX, while in its second part focuses on cardinal characteristics of the lattice and how these relate to some cardinal functions of the space X. [ABSTRACT FROM AUTHOR]

A space X is said to be set star-Lindelöf if for each nonempty subset A of X and each collection U of open sets in X such that Ā⊑⋃u, there is a countable subset V of U such that A ⊑ St(⋃V; u). The class of set star-Lindelöf spaces lie between the class of Lindelöf spaces and the class of star-Lindelöf spaces. In this paper, we investigate the relationship between set star-Lindelöf spaces and other related spaces by providing some suitable examples and study the topological properties of set star-Lindelöf spaces. [ABSTRACT FROM AUTHOR]

In this paper, we study the behavior of some topological and cardinal properties of topological spaces under the influence of the NτΦ-kernel of a space X. It has been proved that the NτΦ-kernel of a space X preserves the density and the network p - weight of normal spaces. Besides, shown that the N-compact kernel of a space X preserves the Souslin properties, the weight, the density, and the p-network weight of normal spaces. [ABSTRACT FROM AUTHOR]

Whenever the closure of an open set is also open, it is called e-open and if a space have a base consisting of e-open sets, it is called e-space. In this paper we first introduce and study e-spaces and e-continuous func-tions (we call a function f from a space X to a space Y an e-continuous at x 2 X if for each open set V containing f(x) there is an e-open set containing x with f(U) C V). We observe that the quasicomponent of each point in a space X is determined by e-continuous functions on X and it is characterized as the largest set containing the point on which every e-continuous function on X is constant. Next, we study the rings Ce(X) of all real valued e-continuous functions on a spaCe X. It turns out that Ce(X) coincides with the ring of real valued clopen continuous functions on X which is a C(Y) for a zero-dimensional spaCe Y whose elements are the quasicomponents of X. Using this fact we characterize the real maximal ideals of Ce(X) and also give a natural representation of its maximal ideals. Finally we have shown that Ce(X) determines the topology of X if and only if it is a zero-dimensional spaCe. [ABSTRACT FROM AUTHOR]

In this paper, we obtain some results on the relationships between different ideal convergence modes namely, IK, IK*, I, K, I [K and (I [K)*. We introduce a topological space namely IK-sequential space and show that the class of IK-sequential spaces contain the sequential spaces. Further IK-notions of cluster points and limit points of a function are also introduced here. For a given sequence in a topological space X, we characterize the set of IK-cluster points of the sequence as closed subsets of X. [ABSTRACT FROM AUTHOR]

c-realcompact spaces are introduced by Karamzadeh and Keshtkar in Quaest. Math. 41, no. 8 (2018), 1135-1167. We offer a characterization of these spaces X via c-stable family of closed sets in X by showing that X is c-realcompact if and only if each c-stable family of closed sets in X with finite intersection property has nonempty intersection. This last condition which makes sense for an arbitrary topological space can be taken as an alternative definition of a c-realcompact space. We show that each topological space can be extended as a dense subspace to a c-realcompact space with some desired extension properties. An allied class of spaces viz CP-compact spaces akin to that of c-realcompact spaces are introduced. The paper ends after examining how far a known class of c-realcompact spaces could be realized as CP-compact for appropriately chosen ideal P of closed sets in X. [ABSTRACT FROM AUTHOR]

One of the fundamental problems in rings of continuous functions is to extract those spaces X for which C(X) determines X, that is to investigate X and Y such that C(X) isomorphic with C(Y ) implies X homeomorphic with Y . Later S. Banach and M. Stone proved independently with slight variance, that if X is a compact Hausdorff space, C(X) also determine X. Their works were maximally extended by E. Hewitt who introduced realcompact spaces and later Melvin Henriksen and Biswajit Mitra solved the problem for locally compact and nearly realcompact spaces. In this paper we tried to develop a unified theory of this problem to cover up all the works in the literature introducing the notion P-compact spaces. [ABSTRACT FROM AUTHOR]

In this paper we continue to research relationships between closure-type properties of hyperspaces over a space X and covering properties of X. For a Hausdorff space X we denote by 2X the family of all closed subsets of X. We investigate selection properties of the bitopological space (2X, Δ1+, Δ2+) where Δi+ is the upper Δi-topology for each i=1,2. [ABSTRACT FROM AUTHOR]

In this paper, closed ideals in Cc(X), the functionally countable subal-gebra of C(X), with the mc-topology are studied. We show that if X is a CUC-space, then C*c (X) with the uniform norm-topology is a Banach algebra. Closed ideals in Cc(X) as a modified countable analogue of closed ideals in C(X) with the m-topology, are characterized. For a zero-dimensional space X, we show that a proper ideal in Cc(X) is closed if and only if it is an intersection of maximal ideals of Cc(X). It is also shown that every ideal in Cc(X) with the mc-topology is closed if and only if X is a P-space if and only if every ideal in C(X) with the m-topology is closed. Also, for a strongly zero-dimensional space X, it is proved that every properly closed ideal in C*c (X) is an intersection of maximal ideals of C*c (X) if and only if X is pseudocompact if and only if every properly closed ideal in C*(X) is an intersection of maximal ideals of C*(X). Finally, we show that if X is a P-space, then the family of ec-ultrafilters and zc-ultrafilter coincide. [ABSTRACT FROM AUTHOR]

Although traditional and idempotent mathematics are "parallel", by an application of the category theory we show that objects obtained the similar rules over traditional and idempotent mathematics must not be "parallel". At first we establish for a compact metric space X the spaces P(X) of probability measures and I(X) idempotent probability measures are homeomorphic ("parallelism"). Then we construct an example which shows that the constructions P and I form distinguished functors from each other ("parallelism" negation). Further for a com-pact Hausdorff space X we establish that the hereditary normality of I3(X)\X implies the metrizability of X. [ABSTRACT FROM AUTHOR]