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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]

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]

Given Tychonoff spaces X and Y, Uspenskij proved in [15] that if X is pseudocompact and Cp(X) is uniformly homeomorphic to Cp(Y), then Y is also pseudocompact. In particular, if Cp(X) is linearly homeomorphic to Cp(Y), then X is pseudocompact if and only if so is Y. This easily implies Arhangel'skii's theorem [1] which states that, in the case when Cp(X) is linearly homeomorphic to Cp(Y the space X is compact if and only if Y is compact. We will establish that existence of a linear homeomorphism between the spaces Cp*(X) and Cp*(Y) implies that X is (pseudo)compact if and only if so is Y. We will also show that the methods of proof used by Arhangel'skii and Uspenskij do not work in our case. [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]

In this paper, we prove the following result. Let X be a complete metric space of weight and be a set such that . Then there is no continuous bijection of the subspace onto a -compact space. As a result, there is no continuous bijection of the subspace onto a Polish space. Thus, it has been proved that metric compact spaces are not -spaces for any uncountable cardinal number . This result answers the question asked by E.G. Pytkeev in his coauthored work "On the properties of subclasses of weakly dyadic compact sets" to be published in the Siberian Mathematical Journal. [ABSTRACT FROM AUTHOR]

Given a functionally countable space X, if K is a scattered Corson compact space, then the product X × K is functionally countable. If the space X is, additionally, weakly ω 1 -Lindelöf, then X × L is functionally countable for any scattered Lindelöf space L. In particular, if X is a weakly Lindelöf functionally countable space, then the product X × L is functionally countable whenever L is a scattered Lindelöf space. We also establish that any finite product of countably compact functionally countable spaces is functionally countable and exponential separability of countably compact spaces X and Y implies that X × Y is exponentially separable if one of them is sequential. If the product X = X 1 × ⋯ × X n of functionally countable spaces X 1 ,... , X n is pseudocompact, then X is functionally countable. [ABSTRACT FROM AUTHOR]

We give a counterexample in this amendment to show that there is an error in consideration of the statement "if f: X → Y and J is an ideal on Y, then f-1(J) = ff-1(J): J ℇ J- is an ideal on X" by Hamlett in his paper "Lindelöf with respect to an ideal" [New Zealand J. Math. 42, 115-120, 2012]. We also modify it here in a new way and henceforth put forward correctly all the results that were based on the said statement derived therein. [ABSTRACT FROM AUTHOR]

We prove a generalized version of Powers' averaging property that characterizes simplicity of reduced crossed products C(X) \rtimes _\lambda G, where G is a countable discrete group, and X is a compact Hausdorff space which G acts on minimally by homeomorphisms. As a consequence, we generalize results of Hartman and Kalantar on unique stationarity to the state space of C(X) \rtimes _\lambda G and to Kawabe's generalized space of amenable subgroups \operatorname {Sub}_a(X,G). This further lets us generalize a result of the first named author and Kalantar on simplicity of intermediate C*-algebras. We prove that if C(Y) \subseteq C(X) is an inclusion of unital commutative G-C*-algebras with X minimal and C(Y) \rtimes _\lambda G simple, then any intermediate C*-algebra A satisfying C(Y) \rtimes _\lambda G \subseteq A \subseteq C(X) \rtimes _\lambda G is simple. [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]

Face presentation attack detection (PAD) has become a clear and present threat for face recognition systems and many countermeasures have been proposed to mitigate it. In these countermeasures, some of them use the features directly extracted from well-known color spaces (e.g., RGB, HSV and YCbCr) to distinguish the fake face images from the genuine ("live") ones. However, the existing color spaces have been originally designed for displaying the visual content of images or videos with high fidelity and are not well suited for directly discriminating the live and fake face images. Therefore, in this paper, we propose a deep-learning system, called CompactNet, for learning a compact space tailored for face PAD. More specifically, the proposed CompactNet does not directly extract the features in existing color spaces, but inputs the color face image into a layer-by-layer progressive space generator. Then, under the optimization of the "points-to-center" triplet loss function, the generator learns a compact space with small intra-class distance, large inter-class distance and a safe interval between different classes. Finally, the feature of the image in compact space is extracted by a pre-trained feature extractor and used for image classification. Reported experiments on three publicly available face PAD databases, namely, the Replay-Attack, the OULU-NPU and the HKBU-MARs V1, show that CompactNet separates very well the two classes of fake and genuine faces and significantly outperforms the state-of-the-art methods for PAD. [ABSTRACT FROM AUTHOR]

As composites of constant, finite (co)product, identity, and powerset functors, Kripke polynomial functors form a relevant class of Set -functors in the theory of coalgebras. The main goal of this paper is to expand the theory of limits in categories of coalgebras of Kripke polynomial functors to the context of quantale-enriched categories. To assume the role of the powerset functor we consider "powerset-like" functors based on the Hausdorff V -category structure. As a starting point, we show that for a lifting of a Set -functor to a topological category X over Set that commutes with the forgetful functor, the corresponding category of coalgebras over X is topological over the category of coalgebras over Set and, therefore, it is "as complete" but cannot be "more complete". Secondly, based on a Cantor-like argument, we observe that Hausdorff functors on categories of quantale-enriched categories do not admit a terminal coalgebra. Finally, in order to overcome these "negative" results, we combine quantale-enriched categories and topology à la Nachbin. Besides studying some basic properties of these categories, we investigate "powerset-like" functors which simultaneously encode the classical Hausdorff metric and Vietoris topology and show that the corresponding categories of coalgebras of "Kripke polynomial" functors are (co)complete. [ABSTRACT FROM AUTHOR]

In this paper, we investigate some topological properties of partial metric spaces (in short PMS). We give some relationship between metric-like PMS, sequentially isosceles PMS and sequentially equilateral PMS. We also prove a type of Urysohn's lemma for metric-like PMS. By applying the construction of Hartman–Mycielski, we show that every bounded PMS can be isometrically embedded into a pathwise connected and locally pathwise connected PMS. In the end, we show that a partial metric space is compact iff it is totally bounded and complete. [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]

A subspace If(X) of the space of idempotent probability measures on a given compact space X is constructed. It is proved that if the initial compact space X is contractible, then If(X) is a contractible compact space as well. It is shown that the shapes of the compact spaces X and If(X) are equal. It is also proved that, given a compact space X, the compact space If(X) is an absolute neighborhood retract if and only if so is X. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring. An ideal I of R is called a d-ideal (f d-ideal) provided that for each a ∈ I (finite subset F of I) and b ∈ R, Ann(a) ⊆ Ann(b) (Ann(F) ⊆ Ann(b)) implies that b ∈ I. It is shown that, the class of z0-ideals (hence all sz0-ideals), maximal ideals in an Artinian or in a Kasch ring, annihilator ideals, and minimal prime ideals over a d-ideal are some distinguished classes of d-ideals. Furthermore, we introduce the class of f d-ideals as a subclass of d-ideals in a commutative ring R. In this regard, it is proved that the ring R is a classical ring with property (A) if and only if every maximal ideal of R is an f d-ideal. The necessary and sufficient condition for which every prime f d-ideal of a ring R being a maximal or a minimal prime ideal is given. Moreover, the rings for which their prime d-ideals are z0-ideals are characterized. Finally, we prove that every prime f d-ideal of a ring R is a minimal prime ideal if and only if for each a ∈ R there exists a finitely generated ideal , for some n ∈ ℕ such that Ann(a, I) = 0. As a consequence, every prime f d-ideal in a reduced ring R is a minimal prime ideal if and only if X= Min(R) is a compact space. [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]

Abstract We present a characterization of spaces of strictly decreasing functions on trees in terms of bisequentiality. This characterization answers Questions 6.1 and 6.2 of [2]. Moreover we study the relation between these spaces and the classes of Corson, Eberlein and uniform Eberlein compacta. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to study the fuzzy structure space of a semiring as well as of a Γ-semiring. We study separation axioms, compactness etc. in the fuzzy structure space of a semiring. Similar study has also been accomplished in the setting of a Γ-semiring S by using the nice interplay between S and its left operator semiring L. [ABSTRACT FROM AUTHOR]

Abstract We prove that generically and modulo a topological conjugacy there is only one dynamical system. [ABSTRACT FROM AUTHOR]

In set theory without the axiom of choice, we investigate the deductive strength of the principle “every topological space with the minimal cover property is compact”, and its relationship with certain notions of finite as well as with properties of linearly ordered sets and partially ordered sets. [ABSTRACT FROM AUTHOR]

Kaplansky [7] proved that C K ( X ) is the intersection of all free maximal ideals in C ( X ) in the case of discrete X , and asked whether the equality holds in general. In this paper we prove that C K ( X ) coincides with the intersection of all free maximal ideals if and only if every open hemicompact z -compact (i.e., every zero-set contained in it is compact) subset of X is relatively compact or equivalently, every open Lindelöf z -compact subset of X is relatively compact. We conclude that the equality holds whenever X is a strongly isocompact space. [ABSTRACT FROM AUTHOR]

We establish that a Tychonoff space X is countable if and only if C p ( X ) is strongly dominated by a second countable space. The same is true for a compact space K such that C p ( K , [ 0 , 1 ] ) is strongly dominated by a second countable space. We also prove that strong domination by a second countable space of the complement of the diagonal of a Tychonoff space X implies that X is an ℵ 0 -space. Our results solve several published open questions. [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]

We prove that if X is a T 1 second countable compact space, then X is a Baire space if and only if every nonempty open subset of X contains a closed subset with nonempty interior. We also prove an analogue of Banach's fixed point theorem for all T 1 compact spaces. Applying the analogue of Banach's fixed point theorem we prove the existence of unique attractors for so called contractive iterated function systems whose Hutchinson operators are closed in compact T 1 spaces. [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]

The Bishop property (♗), introduced recently by K. P. Hart, T. Kochanek and the first-named author, was motivated by Pełczyński’s classical work on weakly compact operators on $C(K)$-spaces. This property asserts that certain chains of functions in said spaces, with respect to a particular partial ordering, must be countable. There are two versions of (♗): one applies to linear operators on $C(K)$-spaces and the other to the compact Hausdorff spaces themselves. We answer two questions that arose after (♗) was first introduced. We show that if $\mathscr{D}$ is a class of compact spaces that is preserved when taking closed subspaces and Hausdorff quotients, and which contains no nonmetrizable linearly ordered space, then every member of $\mathscr{D}$ has (♗). Examples of such classes include all $K$ for which $C(K)$ is Lindelöf in the topology of pointwise convergence (for instance, all Corson compact spaces) and the class of Gruenhage compact spaces. We also show that the set of operators on a $C(K)$-space satisfying (♗) does not form a right ideal in $\mathscr{B}(C(K))$. Some results regarding local connectedness are also presented. [ABSTRACT FROM AUTHOR]

W denotes the category of archimedean ℓ-groups with designated weak unit and ℓ-homomorphisms that preserve the weak unit, and B is the bounded coreflection in W. Comp denotes the category of compact Hausdorff spaces with continuous maps, and Y : W → Comp is the familiar Yosida functor. The enormous collection hcW of hull classes in W and the somewhat less enormous collection ccComp of covering classes in Comp are clearly related 'via' Y, but rather unclearly in the details. In an earlier paper we investigated the relationship between hcW and ccComp and continue to do so here, now focusing on the role of B. Among other things, (i) we define B-saturated hull classes and the sub-species Y-saturated and type μ, (ii) show that for a hull class H of the latter two types, but not always the first, Y[ H] is a covering class, and (iii) describe the various ways the three types relate. This paper is the second installment in our ongoing investigation of the complex taxonomy of hull classes. [ABSTRACT FROM AUTHOR]

In this note we partially answer a question of Cascales, Orihuela and Tkachuk [‘Domination by second countable spaces and Lindelöf ${\rm\Sigma}$-property’, Topology Appl.158(2) (2011), 204–214] by proving that under $CH$ a compact space $X$ is metrisable provided $X^{2}\setminus {\rm\Delta}$ can be covered by a family of compact sets $\{K_{f}:f\in {\it\omega}^{{\it\omega}}\}$ such that $K_{f}\subset K_{h}$ whenever $f\leq h$ coordinatewise. [ABSTRACT FROM AUTHOR]

We study embeddings in maximal pseudocompact spaces together with maximal countable compactness in the class of Tychonoff spaces. It is proved that under MA $${+\neg}$$ CH any compact space of weight $${\kappa < \mathfrak{c}}$$ is a retract of a compact maximal pseudocompact space. If κ is strictly smaller than the first weakly inaccessible cardinal, then the Tychonoff cube [0, 1] is maximal countably compact. However, for a measurable cardinal κ, the Tychonoff cube of weight κ is not even embeddable in a maximal countably compact space. We also show that if X is a maximal countably compact space, then the functional tightness of X is countable. It is independent of ZFC whether every compact space of countable tightness must be maximal countably compact. On the other hand, any countably compact space X with the Mazur property ( $${\equiv}$$ every real-valued sequentially continuous function on X is continuous) must be maximal countably compact. We prove that for any ω-monolithic compact space X, if C( X) has the Mazur property, then it is a Fréchet-Urysohn space. [ABSTRACT FROM AUTHOR]

We develop the construction of fermionic fields in terms of bosonic ones to describe free and interaction models in the circle, using thermofielddynamics. The description in the case of finite temperature is developed for both normal modes and zero modes. The treatment extends the thermofield-bosonization for periodic space. [ABSTRACT FROM AUTHOR]

We study the Bishop–Phelps–Bollobás property for numerical radius within the framework of spaces. We present several sufficient conditions on a compact space K ensuring that has the Bishop–Phelps–Bollobás property for numerical radius. In particular, we show that has such property whenever K is metrizable. [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]

We prove a common fixed point theorem for four two pairs of hybrid mappings in compact metric space satisfying an implicit relations using the concept of strict occasionally weak compatibility which generalize theorems of [1, 4, 7, 28]. As an application we obtain a general fixed point theorem for hybrid pairs satisfying a contractive condition of integral type, which is a new result in compact metric spaces. [ABSTRACT FROM AUTHOR]

Let be a Fréchet space, i.e. a metrizable and complete locally convex space (lcs), its strong second dual with a defining sequence of seminorms induced by a decreasing basis of absolutely convex neighbourhoods of zero , and let be a bounded set. Let be the “worst” distance of the set of weak -cluster points in of sequences in to , and the worst distance of the weak -closure in the bidual of to , where means the natural metric of . Let , provided the involved limits exist. We extend a recent result of Angosto–Cascales to Fréchet spaces by showing that: If , there is a sequence in such that for each -cluster point of and . Moreover, iff . This provides a quantitative version of the weak angelicity in a Fréchet space. Also we show that , where is relatively compact and is the space of -valued continuous functions for a web-compact space and a separable metric space , being now the “worst” distance of the set of cluster points in of sequences in to , respect to the standard supremum metric , and . This yields a quantitative version of Orihuela’s angelic theorem. If is strongly web-compact then ; this happens if for (for instance, if is a (DF)-space or an (LF)-space). In the particular case that is a separable metrizable locally convex space then for each bounded . [ABSTRACT FROM AUTHOR]

Abstract: The classical result of Landau on the existence of kings in finite tournaments (= finite directed complete graphs) is extended to continuous tournaments for which the set X of players is a compact Hausdorff space. The following partial converse is proved as well. Let X be a Tychonoff space which is either zero-dimensional or locally connected or pseudocompact or linearly ordered. If X admits at least one continuous tournament and each continuous tournament on X has a king, then X must be compact. We show that a complete reversal of our theorem is impossible, by giving an example of a dense connected subspace Y of the unit square admitting precisely two continuous tournaments both of which have a king, yet Y is not even analytic (much less compact). [Copyright &y& Elsevier]

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: We show that every abelian topological group contains many interesting sets which are both compact and sequentially compact. Then we can deduce some useful facts, e.g., [(1)] if G is a Hausdorff abelian topological group and is countably additive, then the range is compact metrizable; [(2)] if X is a Hausdorff locally convex space and , then is relatively compact in if and only if F is relatively compact in X, and if and only if F is relatively compact in where is the Dierolf topology which is the strongest -polar topology having the same subseries convergent series as the weak topology. [Copyright &y& Elsevier]

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: Call a space X (weakly) Japanese at a point if X has a closure-preserving local base (or quasi-base respectively) at the point x. The space X is (weakly) Japanese if it is (weakly) Japanese at every . We prove, in particular, that any generalized ordered space is Japanese and that the property of being (weakly) Japanese is preserved by σ-products; besides, a dyadic compact space is weakly Japanese if and only if it is metrizable. It turns out that every scattered Corson compact space is Japanese while there exist even Eberlein compact spaces which are not weakly Japanese. We show that a continuous image of a compact first countable space can fail to be weakly Japanese so the (weak) Japanese property is not preserved by perfect maps. Another interesting property of Japanese spaces is their tightness-monolithity, i.e., in every weakly Japanese space X we have for any set . [Copyright &y& Elsevier]

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]

Let U be a locally compact Hausdorff space that is not compact. Let L(U) denote the family of continuous real valued functions on U such that for each f ∈ L(U) there is a nonzero number p (depending on f) for which f--p vanishes at infinity. Then L(U) is obviously a lattice under the usual ordering of functions. In this paper we prove that L(U), as a lattice alone, characterizes the locally compact space U. Let S be a locally compact Hausdorff space. Define T(S) to be L(S) if S is not compact, and T(S) to be C(S) if S is compact. We prove that any locally compact Hausdorff spaces Sl and S2 are homeomorphic if and only if their associated lattices T(S1) and T(S2) are isomorphic. [ABSTRACT FROM AUTHOR]

Abstract: A space is called d-separable if it has a dense subset representable as the union of countably many discrete subsets. We answer several problems raised by V.V. Tkachuk by showing that [(1)] is d-separable for every space X; [(2)] if X is compact Hausdorff then is d-separable; [(3)] there is a 0-dimensional space X such that is d-separable but (and hence ) is not; [(4)] there is a 0-dimensional space X such that is not d-separable. The proof of (2) uses the following new result: If X is compact Hausdorff then its square has a discrete subspace of cardinality . [Copyright &y& Elsevier]

Abstract: The author presents five popular topics in -theory giving their general description together with some unsolved problems. The main idea is to show the areas of major activities displaying what is being done nowadays and trying to outline the trends of their future development. A big percentage of the cited results and problems are new, i.e., published/obtained in the 21-st century. However, some classical old theorems and questions are also cited the author being convinced that they are worth to be repeated with a new emphasis due to what is achieved at the present time. [Copyright &y& Elsevier]