### ${\aleph}_{0}$-spaces and images of separable metric spaces.

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For every topological property $\mathcal{P}$, we define the class of $\mathcal{P}$-approximable spaces which consists of spaces X having a countable closed cover $\gamma $ such that the “section” $X(x,\gamma )=\bigcap \{F\in \gamma :x\in F\}$ has the property $\mathcal{P}$ for each $x\in X$. It is shown that every $\mathcal{P}$-approximable compact space has $\mathcal{P}$, if $\mathcal{P}$ is one of the following properties: countable tightness, ${\aleph}_{0}$-scatteredness with respect to character, $C$-closedness, sequentiality (the last holds under MA or ${2}^{{\aleph}_{0}}<{2}^{{\aleph}_{1}}$). Metrizable-approximable spaces are studied: every compact space in this class has...

In the present paper, we introduce and study the concept of $\partial $-closed sets in biclosure spaces and investigate its behavior. We also introduce and study the concept of $\partial $-continuous maps.

A metric space (X,d) is monotone if there is a linear order < on X and a constant c such that d(x,y) ≤ cd(x,z) for all x < y < z in X, and σ-monotone if it is a countable union of monotone subspaces. A planar set homeomorphic to the Cantor set that is not σ-monotone is constructed and investigated. It follows that there is a metric on a Cantor set that is not σ-monotone. This answers a question raised by the second author.

We introduce the concept of firm classes of morphisms as basis for the axiomatic study of completions of objects in arbitrary categories. Results on objects injective with respect to given morphism classes are included. In a finitely well-complete category, firm classes are precisely the coessential first factors of morphism factorization structures.

The paper is devoted to the study of the ordered set $A\mathcal{K}\left(X,\alpha \right)$ of all, up to equivalence, $A$-compactifications of an Alexandroff space $\left(X,\alpha \right)$. The notion of $A$-weight (denoted by $aw\left(X,\alpha \right)$) of an Alexandroff space $\left(X,\alpha \right)$ is introduced and investigated. Using results in ([7]) and ([5]), lattice properties of $A\mathcal{K}\left(X,\alpha \right)$ and $A{\mathcal{K}}_{\alpha \mathcal{w}}\left(X,\alpha \right)$ are studied, where $A{\mathcal{K}}_{\alpha \mathcal{w}}\left(X,\alpha \right)$ is the set of all, up to equivalence, $A$-compactifications $Y$ of $\left(X,\alpha \right)$ for which $w\left(Y\right)=aw\left(X,\alpha \right)$. A characterization of the families of bounded functions generating an $A$-compactification of $\left(X,\alpha \right)$ is obtained. The notion...