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We characterize coset spaces of topological groups which are coset spaces of (separable) metrizable groups and complete metrizable (Polish) groups. Besides, it is shown that for a $G$-space $X$ with a $d$-open action there is a topological group $H$ of weight and cardinality less than or equal to the weight of $X$ such that $H$ admits a $d$-open action on $X$. This is further applied to show that if $X$ is a separable metrizable coset space then $X$ has a Polish extension which is a coset space of a Polish group.

A topological space $X$ is defined to have a neighborhood $P$-base at any $x\in X$ from some poset $P$ if there exists a neighborhood base $(U_p[x])_{p\in P}$ at $x$ such that $U_p[x]\subseteq U_{p'}[x]$ for all $p\geq p'$ in $P$. We prove that a compact space is countable, hence metrizable, if it has countable scattered height and a $\mathcal{K}(M)$-base for some separable metric space $M$. This gives a positive answer to Problem 8.6.8 in \cite{Banakh2019}. Let $A(X)$ be the free Abelian topological group on $X$. It is shown that if $Y$ is a retract of $X$ such that the free Abelian topological group $A(Y)$ has a $P$-base and $A(X/Y)$ has a $Q$-base, then $A(X)$ has a $P\times Q$-base. Also if $Y$ is a closed subspace of $X$ and $A(X)$ has a $P$-base, then $A(X/Y)$ has a $P$-base. It is shown that any Freche-Urysohn topological group with a $\mathcal{K}(M)$-base for some separable metric space $M$ is first-countable, hence metrizable. And if $P$ is a poset with calibre~$(\omega_1, \omega)$ and $G$ is a topological group with a $P$-base, then any precompact subset in G is metrizable, hence $G$ is strictly angelic. Applications in function spaces $C_p(X)$ and $C_k(X)$ are discussed. We also give an example of a topological Boolean group of character $\leq \mathfrak{d}$ such that the precompact subsets are metrizable but $G$ doesn't have an $\omega^\omega$-base if $\omega_1

so-metrizable spaces are a class of important generalized metric spaces between metric spaces and s n sn -metrizable spaces where a space is called an so-metrizable space if it has a σ \sigma -locally finite so-network. As the further work that attaches to the celebrated Alexandrov conjecture, it is interesting to characterize so-metrizable spaces by images of metric spaces. This paper gives such characterizations for so-metrizable spaces. More precisely, this paper introduces so-open mappings and uses the “Pomomarev’s method” to prove that a space X X is an so-metrizable space if and only if it is an so-open, compact-covering, σ \sigma -image of a metric space, if and only if it is an so-open, σ \sigma -image of a metric space. In addition, it is shown that so-open mapping is a simplified form of s n sn -open mapping (resp. 2-sequence-covering mapping if the domain is metrizable). Results of this paper give some new characterizations of so-metrizable spaces and establish some equivalent relations among so-open mapping, s n sn -open mapping and 2-sequence-covering mapping, which further enrich and deepen generalized metric space theory.

It is known that if $f\colon {\mathbb R}^2 \to {\mathbb R}$ is a polynomial in each variable, then $f$ is a polynomial. We present generalizations of this fact, when ${\mathbb R}^2$ is replaced by $G\times H$, where $G$ and $H$ are topological Abelian groups. We show, e.g., that the conclusion holds (with generalized polynomials in place of polynomials) if $G$ is a connected Baire space and $H$ has a dense subgroup of finite rank or, for continuous functions, if $G$ and $H$ are connected Baire spaces. The condition of continuity can be omitted if $G$ and $H$ are locally compact or complete metric spaces. We present several examples showing that the results are not far from being optimal., Comment: 15 pages

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 $$\mathbf {ZF}$$ ZF , some are shown to be independent of $$\mathbf {ZF}$$ ZF . For independence results, distinct models of $$\mathbf {ZF}$$ ZF and permutation models of $$\mathbf {ZFA}$$ ZFA with transfer theorems of Pincus are applied. New symmetric models of $$\mathbf {ZF}$$ ZF are constructed in each of which the power set of $$\mathbb {R}$$ 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 $$[0, 1]^{\mathbb {R}}$$ [ 0 , 1 ] R .

Some results between the properties of strongly topological gyrogroups and the properties of their remainders are established. In particular, if a strongly topological gyrogroup G is non-locally compact and G has a first-countable remainder, then $$\chi (G)\le \omega _{1}$$ , $$\omega (G)\le 2^{\omega }$$ and $$|bG|\le 2^{\omega _{1}}$$ . Moreover, it is proved that the property of paracompact p-space of a strongly topological gyrogroup G is equivalent with G having a Lindelof remainder in a compactification. By this result, we prove that if H is a dense subspace of a strongly topological gyrogroup G which is locally pseudocompact and not locally compact, then every remainder of H is pseudocompact. Furthermore, if a strongly topological gyrogroup G has countable pseudocharacter and G is non-metrizable, then all remainders of G are pseudocompact. These two results give partial answers to a question posed by Arhangel’ skiǐ and Choban, see (Topol Appl 157:789–799, 2010, Problem 5.1). Finally, it is shown that the Lindelof property of a non-locally compact strongly topological gyrogroup G is equivalent with having a remainder with subcountable type for some compactifications of G.

A locally convex space (lcs) E is said to have an $$\omega ^{\omega }$$ ω ω -base if E has a neighborhood base $$\{U_{\alpha }:\alpha \in \omega ^\omega \}$$ { U α : α ∈ ω ω } at zero such that $$U_{\beta }\subseteq U_{\alpha }$$ U β ⊆ U α for all $$\alpha \le \beta $$ α ≤ β . The class of lcs with an $$\omega ^{\omega }$$ ω ω -base is large, among others contains all (LM)-spaces (hence (LF)-spaces), strong duals of distinguished Fréchet lcs (hence spaces of distributions $$D^{\prime }(\Omega )$$ D ′ ( Ω ) ). A remarkable result of Cascales-Orihuela states that every compact set in an lcs with an $$\omega ^{\omega }$$ ω ω -base is metrizable. Our main result shows that every uncountable-dimensional lcs with an $$\omega ^{\omega }$$ ω ω -base contains an infinite-dimensional metrizable compact subset. On the other hand, the countable-dimensional vector space $$\varphi $$ φ endowed with the finest locally convex topology has an $$\omega ^\omega $$ ω ω -base but contains no infinite-dimensional compact subsets. It turns out that $$\varphi $$ φ is a unique infinite-dimensional locally convex space which is a $$k_{\mathbb {R}}$$ k R -space containing no infinite-dimensional compact subsets. Applications to spaces $$C_{p}(X)$$ C p ( X ) are provided.

We prove that the Niemytzki plane is $\varkappa$-metrizable and we try to explain the differences between the concepts of a stratifiable space and a $\varkappa$-metrizable space. Also, we give a characterisation of $\varkappa$-metrizable spaces which is modelled on the version described by Chigogidze.

Let $G$ and $H$ be locally compact groups with fixed two-side-invariant Haar measures. A polyhomomorphism $G\to H$ is a closed subgroup $R\subset G\times H$ with a fixed Haar measure, whose marginals on $G$ and $H$ are dominated by the Haar measures on $G$ and $H$. A polyhomomorphism can be regarded as a multi-valued map sending points to sets equipped with 'uniform' measures. For polyhomomorphsisms $G\to H$, $H\to K$ there is a well-defined product $G\to K$. The set of polyhomomorphisms $G\to H$ is a metrizable compact space with respect to the Chabauty--Bourbaki topology and the product is separately continuous. A polyhomomorphism $G\to H$ determines a canonical operator $L^2(H)\to L^2(G)$, which is a partial isometry up to scalar factor. As an example, we consider locally compact infinite-dimensional linear spaces over finite fields and examine closures of groups of linear operators in semigroups of polyendomorphisms., Comment: 26pp, revised version

We prove that every homogeneous countable dense homogeneous topological space containing a copy of the Cantor set is a Baire space. In particular, every countable dense homogeneous topological vector space is a Baire space. It follows that, for any nondiscrete metrizable space $X$, the function space $C_p(X)$ is not countable dense homogeneous. This answers a question posed recently by R. Hern\'andez-Guti\'errez. We also conclude that, for any infinite dimensional Banach space $E$ (dual Banach space $E^\ast$), the space $E$ equipped with the weak topology ($E^\ast$ with the weak$^\ast$ topology) is not countable dense homogeneous. We generalize some results of Hru\v{s}\'ak, Zamora Avil\'es, and Hern\'andez-Guti\'errez concerning countable dense homogeneous products., Comment: slightly modified and expanded version

Answering a question of Uspenskij, we prove that if X is a closed manifold of dimension 2 or higher or the Hilbert cube, then the universal minimal flow of $${\text {Homeo}}(X)$$ is not metrizable. In dimension 3 or higher, we also show that the minimal $${\text {Homeo}}(X)$$ -flow consisting of all maximal, connected chains in X has meager orbits.

In this article we define a metrizable space of multivalued maps. We show that the metric defined in this space is closely related to the homotopy of multivalued maps. Moreover, we study properties of this space and give a few practical applications of the new metric.

Roitman's combinatorial principle $\Delta$ is equivalent to monotone normality of the nabla product, $\nabla (\omega +1)^\omega$. If $\{ X_n : n\in \omega\}$ is a family of metrizable spaces and $\nabla_n X_n$ is monotonically normal, then $\nabla_n X_n$ is hereditarily paracompact. Hence, if $\Delta$ holds then the box product $\square (\omega +1)^\omega$ is paracompact. Large fragments of $\Delta$ hold in $\mathsf{ZFC}$, yielding large subspaces of $\nabla (\omega+1)^\omega$ that are `really' monotonically normal. Countable nabla products of metrizable spaces which are respectively: arbitrary, of size $\le \mathfrak{c}$, or separable, are monotonically normal under respectively: $\mathfrak{b}=\mathfrak{d}$, $\mathfrak{d}=\mathfrak{c}$ or the Model Hypothesis. It is consistent and independent that $\nabla A(\omega_1)^\omega$ and $\nabla (\omega_1+1)^\omega$ are hereditarily normal (or hereditarily paracompact, or monotonically normal). In $\mathsf{ZFC}$ neither $\nabla A(\omega_2)^\omega$ nor $\nabla (\omega_2+1)^\omega$ is hereditarily normal.

Nonconvex optimal-control problems governed by evolution problems in infinite-dimensional spaces (as e.g. parabolic boundary-value problems) needs a continuous (and possibly also smooth) extension on some (preferably convex) compactification, called relaxation, to guarantee existence of their solutions and to facilitate analysis by relatively conventional tools. When the control is valued in some subsets of Lebesgue spaces, the usual extensions are either too coarse (allowing in fact only very restricted nonlinearities) or too fine (being nonmetrizable). To overcome these drawbacks, a compromising convex compactification is here devised, combining classical techniques for Young measures with Choquet theory. This is applied to parabolic optimal control problems as far as existence and optimality conditions concerns.

Let v be a weight on a domain D in a metrizable locally convex space E and F be a complete locally convex space. Denote by Hv(D, F) the weighted space of F-valued holomorphic functions on D satisfying that v.f is bounded, Av(D) subspace of $H_{v}(D, \mathbb C)$ with the unit ball is compact for the open-compact topology. The aim of this paper is to study linearization theorems and approximation properties in several different topologies for weighted spaces Av(D, F) of functions f ∈ Hv(D, F) such that u ∘ f ∈ Av(D) for every continuous linear functional u on F.

Let G be an infinite discrete countable amenable group acting continuously on a compact metrizable space X and Ω be a sequence in G. By using open covers of X, a topological invariant, the topological sequence entropy, is defined. It is shown that the topological sequence entropy can also be defined through relative spanning set or relative separate set. We prove that the topological sequence entropy equals the topological entropy multiplying a constant K ( Ω ) depending only on sequence Ω. The notion of the measure-theoretic sequence entropy is defined. The two sequence entropies are related by the variational principle. The mean topological sequence dimension is defined. It will be nonzero only if the topological sequence entropy is infinite.

We prove the Decomposability Conjecture for functions of Baire class $2$ from a Polish space to a separable metrizable space. This partially answers an important open problem in descriptive set theory.

An ordered topological vector space has the countable dominated extension property if any linear operator ranging in this space, defined on a subspace of a separable metrizable topological vector space, and dominated there by a continuous sublinear operator admits extension to the entire space with preservation of linearity and domination. Our main result is that the strong $$\sigma$$ -interpolation property is a necessary and sufficient condition for a sequentially complete topological vector space ordered by a closed normal reproducing cone to have the countable dominated extension property. Moreover, this fact can be proved in Zermelo–Fraenkel set theory with the axiom of countable choice.

We prove that groups for which every countable subgroup is free (ℵ1-free groups) are n-slender, cm-slender, and lcH-slender. In particular, every homomorphism from a completely metrizable group to an ℵ1-free group has an open kernel. We also show that ℵ1-free abelian groups are lcHslender, which is especially interesting in light of the fact that some ℵ1-free abelian groups are neither n-slender nor cm-slender. The strongly ℵ1-free abelian groups are shown to be n-slender, cm-slender, and lcH-slender. We also give a characterization of the cm-slender and lcH-slender abelian groups.

A new notion of semicontinuity of vector functions in terms of metrizable spaces called external semicontinuity (esc) is introduced. We present some of its topological and geometrical properties an...

A C*-algebra is said to be K K -stable if its nonstable K K -groups are naturally isomorphic to the usual K K -theory groups. We study continuous C ( X ) C(X) -algebras, each of whose fibers are K K -stable. We show that such an algebra is itself K K -stable under the assumption that the underlying space X X is compact, metrizable, and of finite covering dimension.

Let us call a (para)topological group \emph{strongly submetrizable} if it admits a coarser separable metrizable (para)topological group topology. We present a characterization of simply $sm$-factorizable (para)topo\-logical groups by means of continuous real-valued functions. We show that a (para)topo\-logical group $G$ is a simply $sm$-factorizable if and only if for each continuous function $f\colon G\to \mathbb{R}$, one can find a continuous homomorphism $\varphi$ of $G$ onto a strongly submetrizable (para)topological group $H$ and a continuous function $g\colon H\to \mathbb{R}$ such that $f=g\circ\varphi$. This characterization is applied for the study of completions of simply $sm$-factorizable topological groups. We prove that the equalities $\mu{G}=\varrho_\omega{G}=\upsilon{G}$ hold for each Hausdorff simply $sm$-factorizable topological group $G$. This result gives a positive answer to a question posed by Arhangel'skii and Tkachenko in 2018. Also, we consider realcompactifications of simply $sm$-factorizable paratopological groups. It is proved, among other results, that the realcompactification, $\upsilon{G}$, and the Dieudonn\'e completion, $\mu{G}$, of a regular simply $sm$-factorizable paratopological group $G$ coincide and that $\upsilon{G}$ admits the natural structure of paratopological group containing $G$ as a dense subgroup and, furthermore, $\upsilon{G}$ is also simply $sm$-factorizable. Some results in [\emph{Completions of paratopological groups, Monatsh. Math. \textbf{183} (2017), 699--721}] are improved or generalized., Comment: Some problems posed in the old version are answered

We prove that every expansive continuous action with the weak specification property of an amenable group $G$ on a compact Hausdorff space $X$ has the Myhill property, i.e., every pre-injective continuous self-mapping of $X$ commuting with the action of $G$ on $X$ is surjective. This extends a result previously obtained by Hanfeng Li in the case when $X$ is metrizable., We have corrected a few typos and removed part (iii) of Theorem 4.2. It is accepted for publication in the Journal of Dynamical and Control Systems

A compact graph-like space is a triple $(X,V,E)$ where $X$ is a compact, metrizable space, $V \subseteq X$ is a closed zero-dimensional subset, and $E$ is an index set such that $X \setminus V \cong E \times (0,1)$. New characterizations of compact graph-like spaces are given, connecting them to certain classes of continua, and to standard subspaces of Freudenthal compactifications of locally finite graphs. These are applied to characterize Eulerian graph-like compacta.

For a locally compact metrizable group G, we consider the action of $$\mathrm{Aut}(G)$$ on $$\mathrm{Sub}_G$$ , the space of all closed subgroups of G endowed with the Chabauty topology. We study the structure of groups G admitting automorphisms T which act expansively on $$\mathrm{Sub}_G$$ . We show that such a group G is necessarily totally disconnected, T is expansive and that the contraction groups of T and $$T^{-1}$$ are closed and their product is open in G; moreover, if G is compact, then G is finite. We also obtain the structure of the contraction group of such T. For the class of groups G which are finite direct products of $${\mathbb {Q}}_p$$ for distinct primes p, we show that $$T\in \mathrm{Aut}(G)$$ acts expansively on $$\mathrm{Sub}_G$$ if and only if T is expansive. However, any higher dimensional p-adic vector space $${\mathbb {Q}}_p^n$$ , ( $$n\ge 2$$ ), does not admit any automorphism which acts expansively on $$\mathrm{Sub}_G$$ .

Compact subspaces of function spaces with the shift flow figure prominently into the study of dynamical systems where the evolution variable (usually time) is $$\mathbb {R}, \mathbb {Z}, \mathbb {R}^+$$ or $$\mathbb {Z}^+$$ , i.e. continuous or discrete flows or semi-flows. Many important theories depend on this compactness. Up to this point theorems guaranteeing compactness have been limited to three function spaces. These are the functions continuous on $$\mathbb {R}$$ with the topology of uniform convergence on compact subsets of $$\mathbb {R}$$ , the space $$L^p_{loc}(\mathbb {R})$$ with the $$L^p_{loc}(\mathbb {R})$$ topology and finally the non metrizable space $$L^{\infty }_{loc}(\mathbb {R})$$ with an induced weak $$^*$$ topology. In this publication we introduce a new metric that takes e.g. unbounded rapidly oscillatory functions and shows that their hulls are compact in the shift flow. This extends the results obtained in the 1970’s on Lifting Theorems and Spectral Theory by the author and George Sell, as well as any application involving a compact dynamical system.

In [6], given a metrizable profinite group $G$, a cardinal invariant of the continuum $\mathfrak{fm}(G)$ was introduced, and a positive solution to the Haar Measure Problem for $G$ was given under the assumption that $\mathrm{non}(\mathcal{N}) \leq \mathfrak{fm}(G)$. We prove here that it is consistent with ZFC that there is a metrizable profinite group $G_*$ such that $\mathrm{non}(\mathcal{N}) > \mathfrak{fm}(G_*)$, thus demonstrating that the strategy of [6] does not suffice for a general solution to the Haar Measure Problem.

Near-openly generated groups are introduced. They form a topological and multiplicative subclass of -factorizable groups. Dense and open subgroups, quotients and the Raikov completion of a near-openly generated group are near-openly generated. Almost connected pro-Lie groups, Lindelöf almost metrizable groups and the spaces of all continuous real-valued functions on a Tychonoff space with pointwise convergence topology are near-openly generated. We provide characterizations of near-openly generated groups using methods of inverse spectra and topological game theory. Bibliography: 24 titles.

We continue to study cluster sets of a function defined on an open subset of a topological space. In our previous papers we characterized the cluster multifunction of any function and of a continuous function. The next aim was to study cluster sets of a quasi-continuous function. But here we solve a more specific problem on cluster sets of a quasi-locally constant function (i.e., a function that is quasi-continuous with respect to the discrete topology on the range space). We prove that for any open dense subset D of a metrizable space X and a dense subspace Y of a metrizable compact space $${\overline{Y}}$$, every upper continuous compact-valued multifunction $$\Phi :X\setminus D\multimap {\overline{Y}}$$ is the cluster multifunction $${\overline{f}}$$ of some quasi-locally constant function $$f:D\rightarrow Y$$.

In this paper, we consider projective deformation of the geodesic system of Finsler spaces by holonomy invariant functions. Starting with a Finsler spray $S$ and a holonomy invariant function ${\mathcal{P}}$ , we investigate the metrizability property of the projective deformation $\widetilde{S}=S-2\unicode[STIX]{x1D706}{\mathcal{P}}{\mathcal{C}}$ . We prove that for any holonomy invariant nontrivial function ${\mathcal{P}}$ and for almost every value $\unicode[STIX]{x1D706}\in \mathbb{R}$ , such deformation is not Finsler metrizable. We identify the cases where such deformation can lead to a metrizable spray. In these cases, the holonomy invariant function ${\mathcal{P}}$ is necessarily one of the principal curvatures of the geodesic structure.

We prove that all the compact metric spaces are in the closure of the class of full matrix algebras for the quantum Gromov-Hausdorff propinquity. We also show that given an action of a compact metrizable group G on a quasi-Leibniz compact quantum metric space (A,Lip), the function associating any closed subgroup of G group to its fixed point C*-subalgebra in A is continuous from the topology of the Hausdorff distance to the topology induced by the propinquity. Our techniques are inspired from our work on AF algebras as quantum metric spaces, as they are based on the use of various types of conditional expectations., 12 pages. New section and new result on fixed point C*-subalgebra continuity wrt the propinquity. New title and abstract to reflect added content

This article is devoted to the interplay between forcing with fusion and combinatorial covering properties. We discuss known instances of this interplay as well as present a new one, namely that in the Laver model for the consistency of the Borel's conjecture, the Hurewicz property is preserved by finite products of metrizable spaces., Comment: Comments are welcome. To appear in Contemporary Mathematics volume devoted to 100 years of Borel Conjecture. arXiv admin note: text overlap with arXiv:1712.03899

In this thesis we present an overview of some important known facts related to topology, geometry and calculus on metric spaces. We discuss the well known problem of the existence of a lipschitz equivalent metric to a given quasiultrametric, revisiting known results and counterexamples and providing some new theorems, in an unified approach. Also, in the general setting of a quasi-metric doubling space, suitable partition of unity lemmas allows us to obtain, in step two Carnot groups, the well known Whitney’s extension theorem for a given real function of class C^m defined on a closed subset of the whole space: this result relies on relevant properties of the symmetrized Taylor’s polynomial recently introduced in this setting. Finally, some first interesting investigations on Menger convexity in the setting of a general metric spaces concludes this work.

In this paper, we introduce coproducts of proximity spaces. After exploring several of their basic properties, we show that given a collection of proximity spaces, the coproduct of their Smirnov compactifications proximally and densely embeds in the Smirnov compactification of the coproduct of the original proximity spaces. We also show that the dense proximity embedding is a proximity isomorphism if and only if the index set is finite. After constructing a number of examples of coproducts and their Smirnov compactifications, we explore several properties of the Smirnov compactification of the coproduct, including its metrizability, connectedness of the boundary, dimension, and its relation to the Stone-Cech compactification. In particular, we show that the Smirnov compactification of the infinite coproduct is never metrizable and that its boundary is highly disconnected. We also show that the proximity dimension of the Smirnov compactification of the coproduct equals the supremum of the covering dimensions of the individual Smirnov compactifications and that the Smirnov compactification of the coproduct is homeomorphic to the Stone-Cech compactification if and only if each individual proximity space is equipped with the Stone-Cech proximity. We finish with an example of a coproduct with the covering dimension $0$ but the proximity dimension $\infty.$, Comment: 24 pages

Following his discovery that finite metric spaces have injective envelopes naturally admitting a polyhedral structure, Isbell, in his pioneering work on injective metric spaces, attempted a characterization of cellular complexes admitting the structure of an injective metric space. A bit later, Mai and Tang confirmed Isbell's conjecture that a simplicial complex is injectively metrizable if and only if it is collapsible. Considerable advances in the understanding, classification and applications of injective envelopes have since been made by Dress, Huber, Sturmfels and collaborators, and most recently by Lang. Unfortunately a combination theory for injective polyhedra is still unavailable. Here we expose a connection to the duality theory of cubings -- simply connected non-positively curved cubical complexes -- which provides a more principled and accessible approach to Mai and Tang's result, providing one with a powerful tool for systematic construction of locally-compact injective metric spaces: Main Theorem. Any complete pointed Gromov--Hausdorff limit of locally-finite piecewise-$\ell_\infty$ cubings is injective. This result may be construed as a combination theorem for the simplest injective polytopes, $\ell_\infty$-parallelopipeds, where the condition for retaining injectivity is the combinatorial non-positive curvature condition on the complex. Thus it represents a first step towards a more comprehensive combination theory for injective spaces. In addition to setting the earlier work on injectively metrizable complexes within its proper context of non-positively curved geometry, this paper is meant to provide the reader with a systematic review of the results~ ---~ otherwise scattered throughout the geometric group theory literature~ ---~ on the duality theory and the geometry of cubings, which make this connection possible., Significantly expanded and revised. 40 pages, 6 figures, to appear in Expositiones Mathematicae (electronic https://doi.org/10.1016/j.exmath.2018.06.001)

Let $$\{ G_n\}_{n\in \omega }$$ be a closed tower of metrizable groups. Under a mild condition called (GC) and which is strictly weaker than PTA condition introduced by Shimomura et al. (J Math Kyoto Univ 38:551–578, 1998), we show that: (1) the inductive limit $$G=\text{ g- }\underrightarrow{\lim }\,G_n$$ of the tower is a Hausdorff group, (2) every $$G_n$$ is a closed subgroup of G, (3) if K is a compact subset of G, then $$K\subseteq G_m$$ for some $$m\in \omega $$ , (4) G has countable tightness and a $$\mathfrak {G}$$ -base, (5) G is an $$\aleph $$ -space, (6) G is a sequentially Ascoli space if and only if either (i) there is an $$m\in \omega $$ such that $$G_n$$ is open in $$G_{n+1}$$ for every $$n\ge m$$ , so G is metrizable, or (ii) all groups $$G_n$$ are locally compact and G is a sequential non-Frechet–Urysohn space.

[EN] Given a continuous map f : X -> X on a metric space, it induces the maps f over bar :K(X) -> K(X), on the hyperspace of nonempty compact subspaces of X, and (f) over cap :F(X) -> F(X), on the space of normal fuzzy sets, consisting of the upper semicontinuous functions u:X -> [0,1] with compact support. Each of these spaces can be endowed with a respective metric. In this work, we studied the relationships among the dynamical systems (X,f), (K(X),f over bar ), and (F(X),(f) over cap). In particular, we considered several dynamical properties related to chaos: Devaney chaos, A-transitivity, Li-Yorke chaos, and distributional chaos, extending some results in work by Jardon, Sanchez and Sanchis (Mathematics 2020, 8, 1862) and work by Bernardes, Peris and Rodenas (Integr. Equ. Oper. Theory 2017, 88, 451-463). Especial attention is given to the dynamics of (continuous and linear) operators on metrizable topological vector spaces (linear dynamics)., This work was supported by MCIN/AEI/10.13039/501100011033, Project PID2019-105011GB-I00, and by Generalitat Valenciana, Projects PROMETEO/2017/102 and PROMETEU/2021/070.

In this paper, we investigate and explore the properties of quasi-topological loops with respect to irresoluteness. Moreover, we construct an example of a quasi-irresolute topological inverse property-loop by using a zero-dimensional additive metrizable perfect topological inverse property-loop [Formula: see text] with relative topology [Formula: see text].

For a Tychonoff space X, let $$C_k(X)$$ and $$C_p(X)$$ be the spaces of real-valued continuous functions C(X) on X endowed with the compact-open topology and the pointwise topology, respectively. If X is compact, the classic result of A. Grothendieck states that $$C_k(X)$$ has the Dunford–Pettis property and the sequential Dunford–Pettis property. We extend Grothendieck’s result by showing that $$C_k(X)$$ has both the Dunford–Pettis property and the sequential Dunford–Pettis property if X satisfies one of the following conditions: (1) X is a hemicompact space, (2) X is a cosmic space (= a continuous image of a separable metrizable space), (3) X is the ordinal space $$[0,\kappa )$$ for some ordinal $$\kappa $$ , or (4) X is a locally compact paracompact space. We show that if X is a cosmic space, then $$C_k(X)$$ has the Grothendieck property if and only if every functionally bounded subset of X is finite. We prove that $$C_p(X)$$ has the Dunford–Pettis property and the sequential Dunford–Pettis property for every Tychonoff space X, and $$C_p(X) $$ has the Grothendieck property if and only if every functionally bounded subset of X is finite.

A fundamental result proved by Bourgain, Fremlin and Talagrand states that the space $B_1(M)$ of Baire one functions over a Polish space $M$ is an angelic space. Stegall extended this result by showing that the class $B_1(M,E)$ of Baire one functions valued in a normed space $E$ is angelic. These results motivate our study of various topological properties in the classes $B_\alpha(X,G)$ of Baire-$\alpha$ functions, where $\alpha$ is a nonzero countable ordinal, $G$ is a metrizable non-precompact abelian group and $X$ is a $G$-Tychonoff first countable space. In particular, we show that (1) $B_\alpha(X,G)$ is a $\kappa$-Fr\'{e}chet--Urysohn space and hence it is an Ascoli space, and (2) $B_\alpha(X,G)$ is a $k$-space iff $X$ is countable., Comment: arXiv admin note: text overlap with arXiv:1812.10166

In this short note, we give some new results on continuous bounded cohomology groups of topological semigroups with values in complex field. We show that the second continuous bounded cohomology group of a compact metrizable semigroup, is a Banach space. Also, we study cohomology groups of amenable topological semigroups, and we show that cohomology groups of rank greater than one of a compact left or right amenable semigroup, are trivial. Also, we give some examples and applications about topological lattices.

We consider positive linear operators acting on C(K), where K is a metrizable Bauer simplex. For such an operator L we investigate the limit of the iterates L m , when m → ∞ . Qualitative results and rates of convergence are obtained. The general results are illustrated by examples involving classical operators.

Given a separable metrizable h-homogeneous space X, it is proved that X is a $$\textsf{CDH}$$ space with a dense completely metrizable subspace if and only if every countable subset of X is included in a Polish subspace of X. We also show that every such space is $$\omega-\textsf{CDH}$$ .