Related papers: Spaces having a small diagonal
We will introduce the countable separation property (CSP) of Banach spaces X, which is defined as follows: For each subset \mathcal{F} of X^{\ast}, which separates X, there exists a countable separating subset \mathcal{F}_{0} of…
\emph{Scalable spaces} are simply connected compact manifolds or finite complexes whose real cohomology algebra embeds in their algebra of (flat) differential forms. This is a rational homotopy invariant property and all scalable spaces are…
We prove that a compact space is monotonically Sokolov if and only if it is monotonically $\omega$-monolithic. This gives answers to several questions of R. Rojas-Hernandez and V. V. Tkachuk.
We discuss two problems concerning the class Eberlein compacta, i.e., weakly compact subspaces of Banach spaces. The first one deals with preservation of some classes of scattered Eberlein compacta under continuous images. The second one…
A cuf space (set, resp.) is a space (set, resp.) which is a countable union of finite subspaces (subsets, resp.). It is proved in $\mathbf{ZF}$ (with the absence of the axiom of choice) that all countable unions of cuf (denumerable, resp.)…
We show that all finite powers of a Hausdorff space X do not contain uncountable weakly separated subspaces iff there is a c.c.c poset P such that 1_P forces that ``X is a countable union of 0-dimensional subspaces of countable weight.'' We…
In this paper, some features of countably $\alpha$-compact topological spaces are presented and proven. The connection between countably $\alpha$% -compact, Tychonoff, and $\alpha$-Hausdorff spaces is explained. The space is countably…
We construct a ZFC example of a nonmetrizable compact space $K$ such that every totally disconnected closed subspace $L\subseteq K$ is metrizable. In fact, the construction can be arranged so that every nonmetrizable compact subspace may be…
In this paper a generalization of Urysohn's metrization theorem is given for higher cardinals. Namely, it is shown that a topological space with a basis of cardinality at most $|\omega_\mu|$ or smaller is $\omega_\mu$-metrizable if and only…
The aim of this paper is to introduce and give preliminary investigation of T-locally compact spaces. Locally compact and T-locally compact are independent of each other. Every Hausdorff, locally compact space is T-locally compact.…
We give several partial positive answers to a question of Juhasz and Szentmiklossy regarding the minimum number of discrete sets required to cover a compact space. We study the relationship between the size of discrete sets, free sequences…
We show that the statement ``In every separable pseudometric space there is a maximal non-strictly \delta-separated set.'' implies the axiom of choice for countable families of sets. This gives answers to a question of Dybowski and…
The aim of this paper is to introduce the concept of Delta-Compact spaces along with some basic properties of it. Here, we try to establish the behavior of Delta-Compact spaces under the continuous mapping. Finally, we define another…
We investigate the geometry of the family $\cal M$ of isometry classes of compact metric spaces, endowed with the Gromov-Hausdorff metric. We show that sufficiently small neighborhoods of generic finite spaces in the subspace of all finite…
We prove that there is a residual subset of the Gromov-Hausdorff space (i.e. the space of all compact metric spaces up to isometry endowed with the Gromov-Hausdorff distance) whose points enjoy several unexpected properties. In particular,…
In this paper, we firstly discuss the question: Is $l_{2}^{\infty}$ homeomorphic to a rectifiable space or a paratopological group? And then, we mainly discuss locally compact rectifiable spaces, and show that a locally compact and…
We say that a (countably dimensional) topological vector space $X$ is orbital if there is $T\in L(X)$ and a vector $x\in X$ such that $X$ is the linear span of the orbit ${T^nx:n=0,1,...}$. We say that $X$ is strongly orbital if,…
This paper is concerned with conditions under which a metric continuum (a compact connected metric space) contains a non-degenerate chainable continuum.
In the main result of the paper we extend Rosenthal's characterization of Banach spaces with the Schur property by showing that for a quasi-complete locally convex space $E$ whose separable bounded sets are metrizable the following…
We establish some geometrical properties of the space of idempotent probability measures. In particular, for a compact $X$ it is established that if the space $I_{3}(X)\backslash X$ is hereditary normally, then $X$ is metrizable; some…