Some relationships between $1$-sequence-covering maps and weak-open maps or sequence-covering $s$-maps are discussed. These results are used to generalize a result from Lin S., Yan P., Sequence-covering maps of metric spaces, Topology Appl. 109 (2001), 301–314.

In this paper, we prove that a space $X$ is a $g$-metrizable space if and only if $X$ is a weak-open, $\pi$ and $\sigma$-image of a semi-metric space, if and only if $X$ is a strong sequence-covering, quotient, $\pi$ and $mssc$-image of a semi-metric space, where “semi-metric” can not be replaced by “metric”.

For a topological space $X$ let $𝒦\left(X\right)$ be the set of all compact subsets of $X$. The purpose of this paper is to characterize Lindelöf Čech-complete spaces $X$ by means of the sets $𝒦\left(X\right)$. Similar characterizations also hold for Lindelöf locally compact $X$, as well as for countably $K$-determined spaces $X$. Our results extend a classical result of J. Christensen.

The notion of a Hausdorff function is generalized to the concept of H-closed function and the concept of an H-closed extension of a Hausdorff function is developed. Each Hausdorff function is shown to have an H-closed extension.

We study when a topological space has a weaker connected topology. Various sufficient and necessary conditions are given for a space to have a weaker Hausdorff or regular connected topology. It is proved that the property of a space of having a weaker Tychonoff topology is preserved by any of the free topological group functors. Examples are given for non-preservation of this property by “nice” continuous mappings. The requirement that a space have a weaker Tychonoff connected topology is rather...

Given a metric continuum $X$ and a positive integer $n$, $F_n\left(X\right)$ denotes the hyperspace of all nonempty subsets of $X$ with at most $n$ points endowed with the Hausdorff metric. For $K\in F_n\left(X\right)$, $F_n(K,X)$ denotes the set of elements of $F_n\left(X\right)$ containing $K$ and $F_n^K\left(X\right)$ denotes the quotient space obtained from $F_n\left(X\right)$ by shrinking $F_n(K,X)$ to one point set. Given a map $f:X\to Y$ between continua, $f_n:F_n\left(X\right)\to F_n\left(Y\right)$ denotes the induced map defined by $f_n\left(A\right)=f\left(A\right)$. Let $K\in F_n\left(X\right)$, we shall consider the induced map in the natural way $f_{n,K}:F_n^K\left(X\right)\to F_n^{f\left(K\right)}\left(Y\right)$. In this paper we consider the maps $f$, $f_n$, $f_{n,K}$ for some $K\in F_n\left(X\right)$ and $f_{n,K}$ for...