For a cardinal $\alpha$, we say that a subset $B$ of a space $X$ is $C_{\alpha }$-compact in $X$ if for every continuous function $fX\to {ℝ}^{\alpha }$, $f\left[B\right]$ is a compact subset of ${ℝ}^{\alpha }$. If $B$ is a $C$-compact subset of a space $X$, then $\rho (B,X)$ denotes the degree of $C_{\alpha }$-compactness of $B$ in $X$. A space $X$ is called $\alpha$-pseudocompact if $X$ is $C_{\alpha }$-compact into itself. For each cardinal $\alpha$, we give an example of an $\alpha$-pseudocompact space $X$ such that $X\times X$ is not pseudocompact: this answers a question posed by T. Retta in “Some cardinal generalizations of pseudocompactness”...

We present several results related to $L\Sigma \left(\kappa \right)$-spaces where $\kappa$ is a finite cardinal or $\omega$; we consider products and some constructions that lead from spaces of these classes to other spaces of similar classes.

For a Tychonoff space $X$, we will denote by $X_0$ the set of its isolated points and $X_1$ will be equal to $X\setminus X_0$. The symbol $C\left(X\right)$ denotes the space of real-valued continuous functions defined on $X$. $\square {ℝ}^{\kappa }$ is the Cartesian product ${ℝ}^{\kappa }$ with its box topology, and $C_{\square }\left(X\right)$ is $C\left(X\right)$ with the topology inherited from $\square {ℝ}^X$. By $\widehat{C}\left(X_1\right)$ we denote the set $\{f\in C\left(X_1\right):f$ can be continuously extended to all of $X\}$. A space $X$ is almost-$\omega$-resolvable if it can be partitioned by a countable family of subsets in such a way that every non-empty open subset of $X$ has a non-empty...

For a topological property $P$, we say that a space $X$ is star $P$ if for every open cover $𝒰$ of the space $X$ there exists $A\subset X$ such that $st(A,𝒰)=X$. We consider space with star countable extent establishing the relations between the star countable extent property and the properties star Lindelöf and feebly Lindelöf. We describe some classes of spaces in which the star countable extent property is equivalent to either the Lindelöf property or separability. An example is given of a Tychonoff star Lindelöf space with...

We characterize spaces with $\sigma$-$n$-linked bases as specially embedded subspaces of separable spaces, and derive some corollaries, such as the $𝐜$-productivity of the property of having a $\sigma$-linked base.

The problem of Y. Tanaka [10] of characterizing the topologies whose products with each first-countable space are sequential, is solved. The spaces that answer the problem are called strongly sequential spaces in analogy to strongly Fréchet spaces.

By $F_n\left(X\right)$, $n\ge 1$, we denote the $n$-th symmetric product of a metric space $(X,d)$ as the space of the non-empty finite subsets of $X$ with at most $n$ elements endowed with the Hausdorff metric $d_H$. In this paper we shall describe that every isometry from the $n$-th symmetric product $F_n\left(X\right)$ into itself is induced by some isometry from $X$ into itself, where $X$ is either the Euclidean space or the sphere with the usual metrics. Moreover, we study the $n$-th symmetric product of the Euclidean space up to bi-Lipschitz equivalence and...

We consider properties of Tanaka spaces (introduced in Mynard F., More on strongly sequential spaces, Comment. Math. Univ. Carolin. 43 (2002), 525–530), strongly sequential spaces, and weakly sequential spaces. Applications include product theorems for these types of spaces.

In this paper the following two propositions are proved: (a) If $X_{\alpha }$, $\alpha \in A$, is an infinite system of connected spaces such that infinitely many of them are nondegenerated completely Hausdorff topological spaces then the box product ${\square }_{\alpha \in A}{X}_{\alpha }$ can be decomposed into continuum many disjoint nonempty open subsets, in particular, it is disconnected. (b) If $X_{\alpha }$, $\alpha \in A$, is an infinite system of Brown Hausdorff topological spaces then the box product ${\square }_{\alpha \in A}{X}_{\alpha }$ is also Brown Hausdorff, and hence, it is connected. A space is Brown if...

We establish a relation between covering properties (e.gĿindelöf degree) of two standard topological spaces (Lemmas 4 and 5). Some cardinal inequalities follow as easy corollaries.