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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]

For each α-ideal of an almost distributive lattice (ADL) to become a σ-ideal, a set of equivalent conditions is derived, which tends to result in a characterization of generalized Stone ADLs. On an ADL, a one-to-one correspondence is derived between the set of all prime σ-ideals of the ADL and the set of all prime σ-ideals of the quotient ADL. Finally, proved some properties of prime σ-ideals of a normal ADL topologically. [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]

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 Iτ(X) of τ-smooth idempotent probability measures on X and show that the space Iτ(X) is Čech-complete if and only if the given space X is Čech-complete.

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]

A celebrated theorem of Namioka and Phelps (Duke Math J 42:735–750, 1975) says that for a compact space X, the Banach space C(X) is Asplund iff X is scattered. In our paper we extend this result to the space of continuous real-valued functions endowed with the compact-open topology C k (X) for several natural classes of non-compact Tychonoff spaces X. The concept of Δ 1 -spaces introduced recently in Ka̧kol et al. (Some classes of topological spaces extending the class of Δ -spaces, submitted for publication) has been shown to be applicable for this research. w ∗ -binormality of the dual of the Banach space C(X) implies that C(X) is Asplund (Kurka in J Math Anal Appl 371:425–435, 2010). In our paper we prove in particular that for a Corson compact space X the converse is true. We establish a tight relationship between the property of w ∗ -binormality of the dual C (X) ′ and the class of compact Δ -spaces X introduced and explored earlier in Ka̧kol and Leiderman (Proc Am Math Soc Ser B 8:86–99, 2021, 8:267–280, 2021). We find a complete characterization of a compact space X such that the dual C (X) ′ possesses a stronger property called effective w ∗ -binormality. We provide several illustrating examples and pose open questions. [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]

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 (briefly 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 subspace 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, clTZ is a zero-set in T). In 6P.5 of [8] it was shown that a closed countable 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. cozero 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.

The weak topology of a locally convex space (lcs) E is denoted by w. In this paper we undertake a systematic study of those lcs E such that (E, w) is (linearly) Eberlein–Grothendieck (see Definitions 1.2 and 3.1). The following results obtained in our paper play a key role: for every barrelled lcs E, the space (E, w) is Eberlein–Grothendieck (linearly Eberlein–Grothendieck) if and only if E is metrizable (E is normable, respectively). The main applications concern to the space of continuous real-valued functions on a Tychonoff space X endowed with the compact-open topology C k (X) . We prove that (C k (X) , w) is Eberlein–Grothendieck (linearly Eberlein-Grothen—dieck) if and only if X is hemicompact (X is compact, respectively). Besides this, we show that the class of E for which (E, w) is linearly Eberlein–Grothendieck preserves linear continuous quotients. Various illustrating examples are provided. [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]

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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]

Interstage combustion is known for its small axial distance, high combustion efficiency and low lean blowout boundary. However, in a compact space, the distance between the fuel injection location and the wall is confined, and the high-temperature wall may affect the fuel distribution and in turn affect the combustion efficiency. In this study, an experiment of kerosene single-droplet evaporation was conducted, and the fuel distribution in an interstage combustor was numerically simulated. The experimental results showed that the evaporation rate of fuel droplets decreased with increasing distance from the high-temperature wall, and the influence of the lower wall on fuel evaporation was greater than that of the sidewall. The fuel evaporation rate at the high-temperature position within the nonuniform wall temperature field was relatively high. The numerical simulation results indicated that with increasing temperature, the effect of the wall on fuel evaporation increased, as did the uniformity of the fuel distribution. The direction of the temperature gradient imposed the greatest effect on the fuel distribution under nonuniform temperature conditions. This will facilitate the selection of the optimal ignition location within the cavity to achieve efficient combustion. Moreover, it's possible to control the fuel distribution by adjusting the wall temperature. • The kerosene droplet evaporation experiment was conducted in the near-wall area. • The lower wall has a stronger influence on fuel evaporation than the side wall. • Increasing the temperature can improve the uniformity of fuel distribution. • The direction of the temperature gradient has an impact on the fuel distribution. [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]

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]

In the absence of the axiom of choice, the set-theoretic status of many natural statements about metrizable compact spaces is investigated. Some of the statements are provable in ZF , some are shown to be independent of ZF . For independence results, distinct models of ZF and permutation models of ZFA with transfer theorems of Pincus are applied. New symmetric models of ZF are constructed in each of which the power set of R is well-orderable, the Continuum Hypothesis is satisfied but a denumerable family of non-empty finite sets can fail to have a choice function, and a compact metrizable space need not be embeddable into the Tychonoff cube [ 0 , 1 ] R . [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]

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]

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]

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]

In this paper we investigate the existence and stability of quadratic-mean almost periodic mild solutions to stochastic delay functional differential equations driven by fractional Brownian motion with Hurst parameter H > 1/2 , under some suitable assumptions, by means of semigroup of operators and fixed point method.

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]

The video traffic analysis is the most important issue for large scale surveillance. In the large scale surveillance system, huge amount of live digital video data is submitted to the storage servers through the number of externally connected scalable components. The system also contains huge amount of popular and unpopular old videos in the archived storage servers. The video data is delivered to the viewers, partly or completely on demand through a compact system. In real time, huge amount of video data is imported to the viewer's node for various analysis purposes. The viewers use a number of interactive operations during the real time tracking suspect. The compact video on demand system is used in peer to peer mesh type hybrid architecture. The chunk of video objects move fast through the real time generated compact topological space. Video traffic analytics is required to transfer compressed multimedia data efficiently. In this work, we present a dynamically developed topological space, using mixed strategy by game approach to move the video traffic faster. The simulation results are well addressed in real life scenario. [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 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]

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]

We prove that compact Hausdorff spaces with a P -diagonal are metrizable. This answers problem 4.1 (and the equivalent problem 4.12) from Cascales et al. (2011). [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]