We introduce and study the notion of pairwise monotonically normal space as a bitopological extension of the monotonically normal spaces of Heath, Lutzer and Zenor. In particular, we characterize those spaces by using a mixed condition of insertion and extension of real-valued functions. This result generalizes, at the same time improves, a well-known theorem of Heath, Lutzer and Zenor. We also obtain some solutions to the quasi-metrization problem in terms of the pairwise monotone normality.

In this paper, we discuss certain networks on paratopological (or topological) groups and give positive or negative answers to the questions in [Lin2013]. We also prove that a non-locally compact, $k$-gentle paratopological group is metrizable if its remainder (in the Hausdorff compactification) is a Fréchet-Urysohn space with a point-countable cs*-network, which improves some theorems in [Liu C., Metrizability of paratopological $($semitopological$)$ groups, Topology Appl. 159 (2012), 1415–1420], [Liu...

We provide a necessary and sufficient condition for the Higson compactification to be perfect for the noncompact, locally connected, proper metric spaces. We also discuss perfectness of the Smirnov compactification.

In the present article we provide an example of two closed non-$\sigma$-lower porous sets $A,B\subseteq ℝ$ such that the product $A\times B$ is lower porous. On the other hand, we prove the following: Let $X$ and $Y$ be topologically complete metric spaces, let $A\subseteq X$ be a non-$\sigma$-lower porous Suslin set and let $B\subseteq Y$ be a non-$\sigma$-porous Suslin set. Then the product $A\times B$ is non-$\sigma$-lower porous. We also provide a brief summary of some basic properties of lower porosity, including a simple characterization of Suslin non-$\sigma$-lower porous sets in topologically...

If a metrizable space $X$ is dense in a metrizable space $Y$, then $Y$ is called a metric extension of $X$. If $T_1$ and $T_2$ are metric extensions of $X$ and there is a continuous map of $T_2$ into $T_1$ keeping $X$ pointwise fixed, we write $T_1\le T_2$. If $X$ is noncompact and metrizable, then $\left(ℳ\right(X),\le )$ denotes the set of metric extensions of $X$, where $T_1$ and $T_2$ are identified if $T_1\le T_2$ and $T_2\le T_1$, i.e., if there is a homeomorphism of $T_1$ onto $T_2$ keeping $X$ pointwise fixed. $\left(ℳ\right(X),\le )$ is a large complicated poset studied extensively by V. Bel’nov [The structure of...

In this paper we define for fuzzy topological spaces a notion corresponding to proto-metrizable topological spaces. We obtain some properties of these fuzzy topological spaces, particularly we give relations with non-archimedean, and metrizable fuzzy topological spaces.