Let $F\subset C^{\ast }(X)$ be a vector sublattice over $ℝ$ which separates points from closed sets of $X$. The compactification $e_{F}X$ obtained by embedding $X$ in a real cube via the diagonal map, is different, in general, from the Wallman compactification $\omega (Z(F))$. In this paper, it is shown that there exists a lattice $F_z$ containing $F$ such that $\omega (Z(F))=\omega (Z(F_z))=e_{F}X$. In particular this implies that $\omega (Z(F))\ge e_{F}X$. Conditions in order to be $\omega (Z(F))=e_{F}X$ are given. Finally we prove that, if $\alpha X$ is a compactification of $X$ such that $C{l}_{\alpha X}(\alpha X\setminus X)$ is $0$-dimensional, then there is an algebra $A\subset C^{ast}(X)$ such...

If $X$ is a space, then the weak extent$we\left(X\right)$ of $X$ is the cardinal $min\{\alpha :$ If $𝒰$ is an open cover of $X$, then there exists $A\subseteq X$ such that $\left|A\right|=\alpha$ and $St(A,𝒰)=X\}$. In this note, we show that if $X$ is a normal space such that $\left|X\right|=𝔠$ and $we\left(X\right)=\omega$, then $X$ does not have a closed discrete subset of cardinality $𝔠$. We show that this result cannot be strengthened in ZFC to get that the extent of $X$ is smaller than $𝔠$, even if the condition that $we\left(X\right)=\omega$ is replaced by the stronger condition that $X$ is separable.

We show that if a Hausdorff topological space $X$ satisfies one of the following properties: a) $X$ has a countable, discrete dense subset and $X^2$ is hereditarily collectionwise Hausdorff; b) $X$ has a discrete dense subset and admits a countable base; then the existence of a (continuous) weak selection on $X$ implies weak orderability. As a special case of either item a) or b), we obtain the result for every separable metrizable space with a discrete dense subset.

It is proved that for a zero-dimensional space $X$, the function space $C_p(X,2)$ has a Vietoris continuous selection for its hyperspace of at most 2-point sets if and only if $X$ is separable. This provides the complete affirmative solution to a question posed by Tamariz-Mascarúa. It is also obtained that for a strongly zero-dimensional metrizable space $E$, the function space $C_p(X,E)$ is weakly orderable if and only if its hyperspace of at most 2-point sets has a Vietoris continuous selection. This provides a partial...

It is shown that certain weak-base structures on a topological space give a $D$-space. This solves the question by A.V. Arhangel’skii of when quotient images of metric spaces are $D$-spaces. A related result about symmetrizable spaces also answers a question of Arhangel’skii. Theorem.Any symmetrizable space $X$ is a $D$-space $($hereditarily$)$. Hence, quotient mappings, with compact fibers, from metric spaces have a $D$-space image. What about quotient $s$-mappings? Arhangel’skii and Buzyakova have shown that...

In this paper we give a characterization of a separable metrizable space having a metrizable S-weakly infinite-dimensional compactification in terms of a special metric. Moreover, we give two characterizations of a separable metrizable space having a metrizable countable-dimensional compactification.

The main results of this paper are that (1) a space $X$ is $g$-developable if and only if it is a weak-open $\pi$ image of a metric space, one consequence of the result being the correction of an error in the paper of Z. Li and S. Lin; (2) characterizations of weak-open compact images of metric spaces, which is another answer to a question in in the paper of Y. Ikeda, C. liu and Y. Tanaka.