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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]

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]

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]

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]

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]

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]

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 give a finite combinatorial test for finite seminormal functors to possess the property O n and use it in establishing that in some cases this property leads to some well-known functors. For example, if some functor F possesses the property O 2 then F 2 coincides with either exp2 or the squaring functor. Hence we conclude that if F( D ω 1) and D ω 1 are homeomorphic then F 2 is either exp2 or (·)2. [ABSTRACT FROM AUTHOR]

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]