We investigate the solvability in continuous functions of the Abel equation φ(Fx) - φ(x) = 1 where F is a given continuous mapping of a topological space X. This property depends on the dynamics generated by F. The solvability of all linear equations P(x)ψ(Fx) + Q(x)ψ(x) = γ(x) follows from solvability of the Abel equation in case F is a homeomorphism. If F is noninvertible but X is locally compact then such a total solvability is determined by the same property of the cohomological equation φ(Fx)...

Answering a question of Halbeisen we prove (by two different methods) that the algebraic dimension of each infinite-dimensional complete linear metric space X equals the size of X. A topological method gives a bit more: the algebraic dimension of a linear metric space X equals |X| provided the hyperspace K(X) of compact subsets of X is a Baire space. Studying the interplay between Baire properties of a linear metric space X and its hyperspace, we construct a hereditarily Baire linear metric space...

We characterize metric spaces whose hyperspaces of non-empty closed, bounded, compact or finite subsets, endowed with the Attouch-Wets topology, are absolute (neighborhood) retracts.

We construct a consistent example of a normal locally compact metacompact space which is not paracompact, answering a question of A. V. Arkhangel’skiĭ and F. Tall. An interplay between a tower in P(ω)/Fin, an almost disjoint family in ${\left[\omega \right]}^{\omega }$, and a version of an (ω,1)-morass forms the core of the proof. A part of the poset which forces the counterexample can be considered a modification of a poset due to Judah and Shelah for obtaining a Q-set by a countable support iteration.

We show that MA${}_{\sigma -centered}\left({\omega }_1\right)$ implies that normal locally compact metacompact spaces are paracompact, and that MA(${\omega }_1$) implies normal locally compact metalindelöf spaces are paracompact. The latter result answers a question of S. Watson. The first result implies that there is a model of set theory in which all normal locally compact metacompact spaces are paracompact, yet there is a normal locally compact metalindelöf space which is not paracompact.

Let $Con{v}_H\left(X\right)$, $Con{v}_{AW}\left(X\right)$ and $Con{v}_W\left(X\right)$ be the spaces of all non-empty closed convex sets in a normed linear space X admitting the Hausdorff metric topology, the Attouch-Wets topology and the Wijsman topology, respectively. We show that every component of $Con{v}_H\left(X\right)$ and the space $Con{v}_{AW}\left(X\right)$ are AR. In case X is separable, $Con{v}_W\left(X\right)$ is locally path-connected.

Following Banakh and Gabriyelyan (2016) we say that a Tychonoff space X is an Ascoli space if every compact subset of $C_k\left(X\right)$ is evenly continuous; this notion is closely related to the classical Ascoli theorem. Every $k_{ℝ}$-space, hence any k-space, is Ascoli. Let X be a metrizable space. We prove that the space $C_k\left(X\right)$ is Ascoli iff $C_k\left(X\right)$ is a $k_{ℝ}$-space iff X is locally compact. Moreover, $C_k\left(X\right)$ endowed with the weak topology is Ascoli iff X is countable and discrete. Using some basic concepts from probability theory and...

We show that the automorphism group Aut([0,1],λ) of the Lebesgue measure has no non-trivial subgroups of index $<2^{\omega }$.

It is established that a remainder of a non-locally compact topological group $G$ has the Baire property if and only if the space $G$ is not Čech-complete. We also show that if $G$ is a non-locally compact topological group of countable tightness, then either $G$ is submetrizable, or $G$ is the Čech-Stone remainder of an arbitrary remainder $Y$ of $G$. It follows that if $G$ and $H$ are non-submetrizable topological groups of countable tightness such that some remainders of $G$ and $H$ are homeomorphic, then the spaces...

We prove a commutative Gelfand-Naimark type theorem, by showing that the set $C_s\left(X\right)$ of continuous bounded (real or complex valued) functions with separable support on a locally separable metrizable space X (provided with the supremum norm) is a Banach algebra, isometrically isomorphic to C₀(Y) for some unique (up to homeomorphism) locally compact Hausdorff space Y. The space Y, which we explicitly construct as a subspace of the Stone-Čech compactification of X, is countably compact, and if X is non-separable,...

By a dynamical system $(X,T)$ we mean the action of the semigroup $({\mathbb{Z}}^{+},+)$ on a metrizable topological space $X$ induced by a continuous selfmap $T:X\to X$. Let $M\left(X\right)$ denote the set of all compatible metrics on the space $X$. Our main objective is to show that a selfmap $T$ of a compact space $X$ is a Banach contraction relative to some $d_1\in M\left(X\right)$ if and only if there exists some $d_2\in M\left(X\right)$ which, regarded as a $1$-cocycle of the system $(X,T)\times (X,T)$, is a coboundary.