Related papers: A note on Collins-Roscoe Structuring Mechanism

we prove that if $X$ is a locally compact $\sigma$-compact space then on its quotient, $\gamma(X)$ say, determined by the algebra of all real valued bounded continuous functions on $X$, the quotient topology and the completely regular…

The main result of this article is: THEOREM. Every homogeneous locally conical connected separable metric space that is not a $1$-manifold is strongly $n$-homogeneous for each $n \geq 2$ and countable dense homogeneous. Furthermore,…

Let $X$ be a zero-dimensional metric space and $X'$ its derived set. We prove the following assertions: (1) the space $C_k(X,2)$ is an Ascoli space iff $C_k(X,2)$ is $k_\mathbb{R}$-space iff either $X$ is locally compact or $X$ is not…

In this paper we get characterizations countable tightness, countable fan-tightness and countable strong fan-tightness of spaces of quasicontinuous functions with the topology of pointwise convergence from a open Whyburn $T_2$-space $X$…

We work in set-theory without choice ZF. Denoting by AC(N) the countable axiom of choice, we show in ZF+AC(N) that the closed unit ball of a uniformly convex Banach space is compact in the convex topology (an alternative to the weak…

We give new characterizations of spaces $X$ which are $k_\mathbb{R}$-spaces or $s_\mathbb{R}$-spaces. Applying the obtained results we provide some sufficient and necessary conditions on $X$ for which $C_p(X)$ is a $k_\mathbb{R}$-space or…

Whenever P is a proper definable forcing for adding a real, the countable support iteration of P has all the preservation properties it can possibly have, within a wide syntactically identified class of properties.

A space $X$ is sequentially separable if there is a countable $S\subset X$ such that every point of $X$ is the limit of a sequence of points from $S$. In 2004, N.V. Velichko defined and investigated concepts close to sequentially…

We show a necessary and sufficient condition for any ordinal number to be a Polish space. We also prove that for each countable Polish space, there exists a countable ordinal number that is an upper bound for the first component of the…

The paper deals with Ascoli spaces $C_p(X)$ and $C_k(X)$ over Tychonoff spaces $X$. The class of Ascoli spaces $X$, i.e. spaces $X$ for which any compact subset $K$ of $C_k(X)$ is evenly continuous, essentially includes the class of…

We prove that many completeness properties coincide in metric spaces, precompact groups and dense subgroups of products of separable metric groups. We apply these results to function spaces C_p(X,G) of G-valued continuous functions on a…

We studied the relationships between Collins-Roscoe mechanism and D-spaces, proved that well-ordered (F) spaces are D-spaces. This positively answered a question asked by D.Soukup and Y.Xu before.

We prove that the countable product of supercomplete spaces having a countable closed cover consisting of partition-complete subspaces is supercomplete with respect to its metric-fine coreflection. Thus, countable products of…

A space X is finite dimensional, locally compact and separable metrizable if and only if X has a finite basic family: continuous functions Phi_1,...,Phi_n of X to the reals, R, such that for all continuous f from X to R there are g_1,...,…

We construct a compact Hausdorff space $K$ such that the space $P(K)$ of Radon probabiblity measures on $K$ considered with the weak$^*$ topology (induced from the space of continuous functions $C(K)$) is countably tight which is a…

We prove that for each Polish space X, the space C(X) of continuous real-valued functions on X satisfies a strong version of the Pytkeev property, if endowed with the compact-open topology. (This shows that whereas it need not be…

We prove that in the Miller model, every $M$-separable space of the form $C_p(X)$, where $X$ is metrizable and separable, is productively $M$-separable, i.e., $C_p(X)\times Y$ is $M$-separable for every countable $M$-separable $Y$.

A Tychonoff space $X$ is called ({\em sequentially}) {\em Ascoli} if every compact subset (resp. convergent sequence) of $C_k(X)$ is evenly continuous, where $C_k(X)$ denotes the space of all real-valued continuous functions on $X$ endowed…

In this paper we show that forcings which are strongly proper for stationarily many countable elementary submodels preserve each of the following properties of topological spaces: countably tight; Lindel\"of; Rothberger; Menger; and a…

Let $Y$ be a metrizable space containing at least two points, and let $X$ be a $Y_{\mathcal{I}}$-Tychonoff space for some ideal $\mathcal{I}$ of compact sets of $X$. Denote by $C_{\mathcal{I}}(X,Y)$ the space of continuous functions from…