Related papers: On uniformly continuous maps between function spac…

For a topological space $X$ its reflection in a class $\mathsf T$ of topological spaces is a pair $(\mathsf T X,i_X)$ consisting of a space $\mathsf T X\in\mathsf T$ and continuous map $i_X:X\to \mathsf T X$ such that for any continuous map…

We characterize the Hurewicz cofibrations between finite topological spaces, that is, the continuous functions between finite topological spaces that have the homotopy extension property with respect to all topological spaces. In…

We give a unified treatment of the countable dense homogeneity of products of Polish spaces, with a focus on uncountable products. Our main result states that a product of fewer than $\mathfrak{p}$ Polish spaces is countable dense…

We construct a variety of mappings of the unit interval into $\mathcal{L}^p([0,1])$ to generalize classical examples of $\mathcal{L}^p$-convergence of sequences of functions with simultaneous pointwise divergence. By establishing relations…

Let $C(X,E)$ be the linear space of all continuous functions on a compact Hausdorff space $X$ with values in a locally convex space $E$. We characterize maps $T:C(X,E)\to C(Y,E)$ which satisfy $\mathrm{Ran}(TF-TG)\subset\mathrm{Ran}(F-G)$…

Let $P$ be a directed set and $X$ a space. A collection $\mathcal{C}$ of subsets of $X$ is \emph{$P$-locally finite} if $\mathcal{C}=\bigcup \{ \mathcal{C}_p : p \in P\}$ where (i) if $p \le p'$ then $\mathcal{C}_p \subseteq…

Using a factorization theorem due to Pasynkov we provide a short proof of the existence and density of parametric Bing and Krasinkiewicz maps. In particular, the following corollary is established: Let $f\colon X\to Y$ be a surjective map…

In Persistent Homology and Topology, filtrations are usually given by introducing an ordered collection of sets or a continuous function from a topological space to $\R^n$. A natural question arises, whether these approaches are equivalent…

We show that every continuous map from one translationally finite tiling space to another can be approximated by a local map. If two local maps are homotopic, then the homotopy can be chosen so that every interpolating map is also local.

Compact sets in constructive mathematics capture our intuition of what computable subsets of the plane (or any other complete metric space) ought to be. A good representation of compact sets provides an efficient means of creating and…

We present a necessary condition for a pair of $\mathcal{C}(K)$ spaces to be isomorphic in terms of topological properties of Cantor-Bendixon derivatives of $K$. This in particular gives a completely new information about the perfect…

Let $X$ be a topological space. A subset of $C(X)$, the space of continuous real-valued functions on $X$, is a partially ordered set in the pointwise order. Suppose that $X$ and $Y$ are topological spaces, and $A(X)$ and $A(Y)$ are subsets…

For a Hausdorff zero-dimensional topological space $X$ and a totally ordered field $F$ with interval topology, let $C_c(X,F)$ be the ring of all $F-$valued continuous functions on $X$ with countable range. It is proved that if $F$ is either…

We define a new type of Hardy-Orlicz spaces of conformal mappings on the unit disk where in place of the value |f(x)| we consider the intrinsic path distance between f(x) and f(0) in the image domain. We show that if the Orlicz function is…

In this paper, for a metrizable space $Z$, we consider the space of metrics that generate the same topology of $Z$, and that space of metrics is equipped with the supremum metrics. For a metrizable space $X$ and a closed subset $A$ of it,…

We prove, using a weakening of the Proper Forcing Axiom, that any homemomorphism between Cech--Stone remainders of any two locally compact, zero-dimensional Polish spaces is induced by a homeomorphism between their cocompact subspaces.

Let $f\colon X\to Y$ be a perfect surjective map of metrizable spaces. It is shown that if $Y$ is a $C$-space (resp., $\dim Y\leq n$ and $\dim f\leq m$), then the function space $C(X,\uin^{\infty})$ (resp., $C(X,\uin^{2n+1+m})$) equipped…

Let X be a h-homogeneous zero-dimensional compact Hausdorff space, i.e. X is a Stone dual of a homogeneous Boolean algebra. It is shown that the universal minimal space M(G) of the topological group G=Homeo(X), is the space of maximal…

Fix a compact metric space $X$ with finite topological dimension. Let $C^{0}(X)$ be the space of continuous maps on $X$ and $ H^{\alpha}(X)$ the space of $\alpha$-H\"older continuous maps on $X$, for $\alpha\in (0,1].$ $H^{1}(X)$ is the…

For bi-Lipschitz homeomorphisms of a compact manifold it is known that topological entropy is always finite. For compact manifolds of dimension two or greater, we show that in the closure of the space of bi-Lipschitz homeomorphisms, with…