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For the set I (X) of probability measures on a compact Hausdorff space X, we propose a new way to introduce the topology by using the open subsets of the space X. Then, among other things, we give a new proof that for a compact Hausdorff space X, the space I (X) is also a compact Hausdorff space. For a Tychonoff space X, we consider the topological space I τ (X) of τ -smooth idempotent probability measures on X and show that the space I τ (X) is Čech-complete if and only if the given space X is Čech-complete. [ABSTRACT FROM AUTHOR]

Given a topological space X, let UX and ηX: X → UX denote, respectively, the Salbany compactification of X and the compactification map called the Salbany map of X. For every continuous function f: X → Y, there is a continuous function Uf: UX → UY, called the Salbany lift of f, satisfying (Uf) ◦ ηX = ηY ◦ f. If a continuous function f: X → Y has a stably compact codomain Y, then there is a Salbany extension F: UX → Y of f, not necessarily unique, such that F ◦ ηX = f. In this paper, we give a condition on a space such that its Salbany map is open. In particular, we prove that in a class of Hausdorff spaces, the spaces with open Salbany maps are precisely those that are almost discrete. We also investigate openness of the Salbany lift and a Salbany extension of a continuous function. Related to open continuous functions are initial maps as well as nearly open maps. It turns out that the Salbany map of every space is both initial and nearly open. We repeat the procedure done for openness of Salbany maps, Salbany lifts and Salbany extensions to their initiality and nearly openness. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the compactness and cardinality of the space C(X, Y ) of continuous functions from a topological space X to Y equipped with the regular topology. We prove that different forms of compactness, such as sequential compactness, countable compactness, and pseudocompactness, coincide on a subset of C(X, Y ) with regular topology. Moreover, we prove the comparison and coincidence of regular topology with the graph topology on the space C(X, Y ). Furthermore, we examine various cardinal invariants, such as density, character, pseudocharacter, etc., on the space C(X, Y ) equipped with the regular topology. In addition, we define a type of equivalence between X and Y in terms of C(X) and C(Y ) endowed with the regular topology and investigate certain cardinal invariants preserved by this equivalence. [ABSTRACT FROM AUTHOR]

Quillen's notion of small object and the Gabriel-Ulmer notion of finitely presentable or generated object are fundamental in homotopy theory and categorical algebra. Do these notions always lead to rather uninteresting classes of objects in categories of topological spaces, such as all finite discrete spaces, or just the empty space, as the examples and remarks in the existing literature may suggest? This article demonstrates that the establishment of full characterizations of these notions (and some natural variations thereof) in many familiar categories of spaces can be quite challenging and may lead to unexpected surprises. In fact, we show that there are significant differences in this regard even amongst the categories defined by the standard separation axioms, with the T1-separation condition standing out. The findings about these specific categories lead us to insights also when considering rather arbitrary full reflective subcategories of the category of all topological spaces. [ABSTRACT FROM AUTHOR]

In this note, a notion of generalized topological entropy for arbitrary subsets of the space of all sequences in a compact topological space is introduced. It is shown that for a continuous map on a compact space, the generalized topological entropy of the set of all orbits of the map coincides with the classical topological entropy of the map. Some basic properties of this new notion of entropy are considered; among them are the behavior of the entropy with respect to disjoint union, cartesian product, component restriction and dilation, shift mapping, and some continuity properties with respect to Vietoris topology. As an example, it is shown that any self-similar structure of a fractal given by a finite family of contractions gives rise to a notion of intrinsic topological entropy for subsets of the fractal. A generalized notion of Bowen's entropy associated to any increasing sequence of compatible semimetrics on a topological space is introduced and some of its basic properties are considered. As a special case for 1 ≤ p ≤ ∞ , the Bowen p-entropy of sets of sequences of any metric space is introduced. It is shown that the notions of generalized topological entropy and Bowen ∞ -entropy for compact metric spaces coincide. [ABSTRACT FROM AUTHOR]

We consider two natural topologies on the space S (X × Y , Z) of all separately continuous functions defined on the product of two topological spaces X and Y and ranged into a topological or metric space Z. These topologies are the cross-open topology and the cross-uniform topology. We show that these topologies coincides if X and Y are pseudocompacts and Z is a metric space. We prove that a compact space K embeds into S (X × Y , Z) for infinite compacts X , Y and a metrizable space Z ⊇ R if and only if the weight of K is less than the sharp cellularity of both spaces X and Y. [ABSTRACT FROM AUTHOR]

An almost P-space is a topological space in which every zero-set is regular-closed. We introduce a large class of spaces, C-almost P-space (briey CAP-space), consisting of those spaces in which the closure of the interior of every zero-set is a zero-set. In this paper we study CAP-spaces. It is proved that if X is a dense and Z#-embedded sub-space of a space T, then T is CAP if and only if X is a CAP and CRZ-extended in T (i.e, for each regular-closed zero-set Z in X, clT Z is a zero-set in T). In 6P.5 of [8] it was shown that a closed count- able union of zero-sets need not be a zero-set. We call X a CZ-space whenever the closure of any countable union of zero-sets is a zero-set. This class of spaces contains the class of P-spaces, perfectly normal spaces, and is contained in the cozero complemented spaces and CAP-spaces. In this paper we study topological properties of CZ (resp. coz-ero complemented)-space and other classes of topological spaces near to them. Some algebraic and topological equivalent conditions of CZ (resp. cozero complemented)-space are characterized. Examples are provided to illustrate and delimit our results. [ABSTRACT FROM AUTHOR]

As it was introduced by Tkachuk and Wilson in [7], a topological space X is cellular-compact if for any cellular, i.e. disjoint, family U of non-empty open subsets of X there is a compact subspace K ⊂ X such that K ∩ U ≠ ∅ for each U ∈ U . In this note we answer several questions raised in [7] by showing that any first countable cellular-compact T2-space is T3, and so its cardinality is at most c = 2 ω ; cov (M) > ω 1 implies that every first countable and separable cellular-compactT2-space is compact; if there is no S-space then any cellular-compact T3-space of countable spread is compact; M A ω 1 implies that every point of a compact T2-space of countable spread has a disjoint local π -base. [ABSTRACT FROM AUTHOR]

While topology given by a linear order has been extensively studied, this cannot be said about the case when the order is given only locally. The aim of this paper is to fill this gap. We consider relation between local orderability and separation axioms and give characterisation of those regularly locally ordered spaces which are connected, locally connected or Lindelöf. We prove that local orderability is hereditary on open, connected or compact subsets. A collection of interesting examples is also offered. [ABSTRACT FROM AUTHOR]

We introduce some cardinal functions on the product X 1 × ⋯ × X n of topological spaces X k , which are formulated in terms of the type of local finiteness of families of open sets. Using these cardinal functions, we obtain necessary and sufficient conditions that every separately continuous function or strongly separately continuous function f : X 1 × ⋯ × X n → R depends on ℵ coordinates, where every space X k is a strongly countably Čech complete space. [ABSTRACT FROM AUTHOR]

We introduce and study the notions of upper Namioka property, upper Namioka space and upper co-Namioka space which are development of the notions of Namioka property, Namioka space and co-Namioka space on the case of compact-valued mappings. We obtain the following results: the class of upper Namioka spaces consists of Baire spaces with everywhere dense set of isolated points; any subset of a upper co-Namioka compact space is separable; every well-ordered upper co-Namioka compact and every upper co-Namioka compact Valdivia are metrizable; the double arrow space is not upper co-Namioka; there exist a compact-valued mapping F ∈ L U ( X , Y ) defined on the product of Namioka and co-Namioka spaces such that F has not upper Namioka property; if there exists a non-metrizable linearly ordered upper co-Namioka space, then the set of its non-isolated neighbor points contains a subset always of the first category subset; every compact-valued mapping F ∈ L U ( X , Y ) defined on the product of a β - σ ′ -unfavorable space X and a separable linearly ordered compact space Y has the upper Namioka property. [ABSTRACT FROM AUTHOR]

In this paper we investigate Tychonoff spaces which are retracts of paratopological groups. A strong necessary condition for that is the existence of a certain binary operation on the space (called a τ -twister), which was introduced in [2,3] . Some general theorems are established which imply that βω is not a retract of a paratopological group. We also notice, using some deep results of V.V. Uspenskij, that the space ω 1 of countable ordinals is not a retract of any topological group (see Fact 3 ). [ABSTRACT FROM AUTHOR]

A topological space is said to be paranormal if every countable discrete collection of closed sets { D n : n < ω } can be expanded to a locally finite collection of open sets { U n : n < ω } , i.e. D n ⊂ U n , and D m ∩ U n ≠ ∅ iff D m = D n . It is proved that if F is a normal functor F : C o m p → C o m p of degree ≥3 and the space F ( X ) ∖ X is hereditarily paranormal, then the compact space X is metrizable. [ABSTRACT FROM AUTHOR]

We will prove that a Tychonoff space X is analytic if and only if ( X × X ) \ Δ is dominated by a Polish space; here Δ = { ( x , x ) : x ∈ X } is the diagonal of X . This solves two published open questions. We will also establish under CH, that a Tychonoff space X has a countable network whenever ( X × X ) \ Δ is dominated by a second countable space. [ABSTRACT FROM AUTHOR]

Let F ( X ) denote the free topological group over a Tychonoff space X , F n ( X ) denote the subspace of F ( X ) that consists of all words of reduced length ≤ n with respect to the free basis X for every non-negative integer n and E n ( X ) = F n ( X ) ∖ F n − 1 ( X ) for n ≥ 1 . In this paper, we study topological properties of free topological groups in terms of Arens' space S 2 . The following results are obtained. (1) If the free topological group F ( X ) over a Tychonoff space X contains a non-trivial convergent sequence, then F ( X ) contains a closed copy of S 2 , equivalently, F ( X ) contains a closed copy of S ω , which extends [6, Theorem 1.6] . (2) Let X be a topological space and A = { n 1 , . . . , n i , . . . } be an infinite subset of N . If C = ⋃ i ∈ N E n i ( X ) is κ -Fréchet–Urysohn and contains no copy of S 2 , then X is discrete, which improves [15, Proposition 3.5] . (3) If X is a μ -space and F 5 ( X ) is Fréchet–Urysohn, then X is compact or discrete, which improves [15, Theorem 2.4] . At last, a question posed by K. Yamada is partially answered in a shorter alternative way by means of a Tanaka's theorem concerning Arens' space S 2 . [ABSTRACT FROM AUTHOR]

If C p ( X ) is jointly metrizable on compacta, then p ( X ) ≤ ω but ω 1 need not be a caliber of X . If X is either submetrizable or a P -space, then C p ( C p ( X ) ) is jointly metrizable on compacta and, in particular, all compact subsets of C p ( C p ( X ) ) are metrizable. We show that for any dyadic compact X , the space C p ( X ) is jointly metrizable on compacta. Therefore, the JCM property of C p ( X ) for a compact space X does not imply that X is separable. If X is a compact space of countable tightness and C p ( X ) is jointly metrizable on compacta, then it is independent of ZFC whether X must be separable. [ABSTRACT FROM AUTHOR]

We prove that for a compact Hausdorff space X, if λc (X) < w (X) for every infinite cardinal λ < w (X) and λc (X) < cf (w (X)) for every infinite cardinal λ < cf (w (X)), then Tikhonov cube [0,1] w (X) is a continuous image of X, in particular the cardinality of X is just 2 w (X). As an application of this result, we consider elementary submodel spaces and improve Tall's result in [17]. [ABSTRACT FROM AUTHOR]

In this work we study countably z-compact spaces and z-Lindelof spaces. Several new properties of them are given. It is proved that every countably z-compact space is pseuodocompact (a space on which every real valued continuous function is bounded). Spaces which are countably z-compact but not countably compact are given. It is proved that a space is countably z-compact iff every countable z-closed set is compact. Characterizations of countably z-compact and z-Lindelof spaces by multifunctions are given. [ABSTRACT FROM AUTHOR]

The characterization of stably closed maps of topological spaces as the closed maps with compact fibres and the role of the Kuratowski-Mrówka' Theorem in this characterization are being explored in the general context of lax (T, V )-algebras, for a quantale V and a Set-monad T with a lax extension to V -relations. The general results are being applied in standard (topological and metric) and non-standard (labeled graphs) contexts. [ABSTRACT FROM AUTHOR]

Abstract: Given a space M, a family of sets of a space X is ordered by M if { is a compact subset of M} and implies . We study the class of spaces which have compact covers ordered by a second countable space. We prove that a space belongs to if and only if it is a Lindelöf Σ-space. Under , if X is compact and has a compact cover ordered by a Polish space then X is metrizable; here is the diagonal of the space X. Besides, if X is a compact space of countable tightness and belongs to then X is metrizable in ZFC. We also consider the class of spaces X which have a compact cover ordered by a second countable space with the additional property that, for every compact set there exists with . It is a ZFC result that if X is a compact space and belongs to then X is metrizable. We also establish that, under CH, if X is compact and belongs to then X is countable. [ABSTRACT FROM AUTHOR]

Abstract: Answering questions raised by O.T. Alas and R.G. Wilson, or by these two authors together with M.G. Tkachenko and V.V. Tkachuk, we show that every minimal SC space must be sequentially compact, and we produce the following examples: [–] a KC space which cannot be embedded in any compact KC space; [–] a countable KC space which does not admit any coarser compact KC topology; [–] a minimal Hausdorff space which is not a k-space. We also give an example of a compact KC space such that every nonempty open subset of it is dense, even if, as pointed out to us by the referee, a completely different construction carried out by E.K. van Douwen in 1993 leads to a space with the same properties. [Copyright &y& Elsevier]

Some problems arising out of recent work of P.M. Gartside and J.T.H. Lo are here surveyed. [Copyright &y& Elsevier]

We study a new cardinal-valued invariant $ndw(X)$ (calling it the nd-weight of X) of a topological space which is defined as the least upper bound of the weights of nowhere dense subsets of X. The main result is the proof of the inequality hl(X)≤ndw(X) for compact sets without isolated points ((hl is the hereditary Lindelof number). This inequality implies that a compact space without isolated points of countable nd-weight is completely normal. Assuming the continuum hypothesis, we construct an example of a nonmetrizable compact space of countable nd-weight without isolated points. [ABSTRACT FROM AUTHOR]

Abstract: For a compact space K we consider the set of all -points in K, i.e., points with countable base of neighborhoods in K. We show that, for every scattered Eberlein compact space K, the set is a -set in K. We also give an example of a scattered Eberlein compactum with non-metrizable set . Moreover, we give an example of a Corson compact space K such that does not contain any dense -subset of K. This answers three questions of Tkachuk. [Copyright &y& Elsevier]