We prove that, for every finite-dimensional metrizable space, there exists a compactification that is Eberlein compact and preserves both the covering dimension and weight.

A.V. Arkhangel’skii asked that, is it true that every space $Y$ of countable tightness is homeomorphic to a subspace (to a closed subspace) of $C_p\left(X\right)$ where $X$ is Lindelöf? $C_p\left(X\right)$ denotes the space of all continuous real-valued functions on a space $X$ with the topology of pointwise convergence. In this note we show that the two arrows space is a counterexample for the problem by showing that every separable compact linearly ordered topological space is second countable if it is homeomorphic to a subspace of $C_p\left(X\right)$...

We apply elementary substructures to characterize the space $C_p\left(X\right)$ for Corson-compact spaces. As a result, we prove that a compact space $X$ is Corson-compact, if $C_p\left(X\right)$ can be represented as a continuous image of a closed subspace of ${\left(L_{\tau }\right)}^{\omega }\times Z$, where $Z$ is compact and $L_{\tau }$ denotes the canonical Lindelöf space of cardinality $\tau$ with one non-isolated point. This answers a question of Archangelskij [2].

It is shown that no infinite-dimensional Banach space can have a weakly K-analytic Hamel basis. As consequences, (i) no infinite-dimensional weakly analytic separable Banach space E has a Hamel basis C-embedded in E(weak), and (ii) no infinite-dimensional Banach space has a weakly pseudocompact Hamel basis. Among other results, it is also shown that there exist noncomplete normed barrelled spaces with closed discrete Hamel bases of arbitrarily large cardinality.

For a subset $A$ of the real line $ℝ$, Hattori space $H\left(A\right)$ is a topological space whose underlying point set is the reals $ℝ$ and whose topology is defined as follows: points from $A$ are given the usual Euclidean neighborhoods while remaining points are given the neighborhoods of the Sorgenfrey line. In this paper, among other things, we give conditions on $A$ which are sufficient and necessary for $H\left(A\right)$ to be respectively almost Čech-complete, Čech-complete, quasicomplete, Čech-analytic and weakly separated (in...

In this note, we prove that any “bounded” isometries of separable metric spaces can be represented as restrictions of linear isometries of function spaces $C\left(Q\right)$ and $C\left(\Delta \right)$, where $Q$ and $\Delta$ denote the Hilbert cube ${[0,1]}^{\infty }$ and a Cantor set, respectively.

For a non-isolated point $x$ of a topological space $X$ let ${\mathrm{nw}}_{\chi }\left(x\right)$ be the smallest cardinality of a family $𝒩$ of infinite subsets of $X$ such that each neighborhood $O\left(x\right)\subset X$ of $x$ contains a set $N\in 𝒩$. We prove that (a) each infinite compact Hausdorff space $X$ contains a non-isolated point $x$ with ${\mathrm{nw}}_{\chi }\left(x\right)={\aleph }_0$; (b) for each point $x\in X$ with ${\mathrm{nw}}_{\chi }\left(x\right)={\aleph }_0$ there is an injective sequence ${\left(x_n\right)}_{n\in \omega }$ in $X$ that $ℱ$-converges to $x$ for some meager filter $ℱ$ on $\omega$; (c) if a functionally Hausdorff space $X$ contains an $ℱ$-convergent injective sequence for some meager filter...

We construct an example of a Banach space E such that every weakly compact subset of E is bisequential and E contains a weakly compact subset which cannot be embedded in a Hilbert space equipped with the weak topology. This answers a question of Nyikos.

Properties similar to countable fan-tightness are introduced and compared to countable tightness and countable fan-tightness. These properties are also investigated with respect to function spaces and certain classes of continuous mappings.

In [7], M. Levin proved that the set of all Bing maps of a compact metric space to the unit interval is a dense $G_{\delta }$-subset of the space of all maps. In [6], J. Krasinkiewicz independently proved that the set of all Bing maps of a compact metric space to an n-dimensional manifold (n ≥ 1) is a dense $G_{\delta }$-subset of the space of maps. In [9], J. Song and E. D. Tymchatyn, solving some problems of J. Krasinkiewicz ([6]), proved that the set of all Bing maps of a compact metric space to a nondegenerate connected...