We prove that a cosmic space (= a Tychonoff space with a countable network) is analytic if it is an image of a $K$-analytic space under a measurable mapping. We also obtain characterizations of analyticity and $\sigma$-compactness in cosmic spaces in terms of metrizable continuous images. As an application, we show that if $X$ is a separable metrizable space and $Y$ is its dense subspace then the space of restricted continuous functions $C_p(X\mid Y)$ is analytic iff it is a $K_{\sigma \delta }$-space iff $X$ is $\sigma$-compact.

Several remarks on the properties of approximation by points (AP) and weak approximation by points (WAP) are presented. We look in particular at their behavior in product and at their relationships with radiality, pseudoradiality and related concepts. For instance, relevant facts are: (a) There is in ZFC a product of a countable WAP space with a convergent sequence which fails to be WAP. (b) $C_p$ over $\sigma$-compact space is AP. Therefore AP does not imply even pseudoradiality in function spaces, while...

We characterize Peano continua using Bing-Krasinkiewicz-Lelek maps. Also we deal with some topics on Whitney preserving maps.

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.