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...

We investigate properties of the class of compact spaces on which every regular Borel measure is separable. This class will be referred to as MS. We discuss some closure properties of MS, and show that some simply defined compact spaces, such as compact ordered spaces or compact scattered spaces, are in MS. Most of the basic theory for regular measures is true just in ZFC. On the other hand, the existence of a compact ordered scattered space which carries a non-separable (non-regular) Borel measure...

In questo articolo vengono presentate e studiate le nozioni di insieme ${\mathrm{\Lambda }}_{\delta }$ e di insieme $\left(\mathrm{\Lambda },\delta \right)$-chiuso. Inoltre, vengono introdotte le nozioni di $\left(\mathrm{\Lambda },\delta \right)$-continuità, $\left(\mathrm{\Lambda },\delta \right)$-compatezza e $\left(\mathrm{\Lambda },\delta \right)$-connessione e vengono fornite alcune caratterizzazioni degli spazi $\delta -T_0$ e $\delta -T_1$. Infine, viene mostrato che gli spazi $\left(\mathrm{\Lambda },\delta \right)$-connessi e $\left(\mathrm{\Lambda },\delta \right)$-compatti vengono preservati mediante suriezioni $\delta$-continue.

Generalizations of earlier negative results on Property $\left(a\right)$ are proved and two questions on an $\left(a\right)$-version of Jones’ Lemma are posed. We discuss these questions in the realm of locally compact spaces. Using dominating families of functions as a tool, we prove that under the assumptions “$2^{\omega }$ is regular” and “$2^{\omega }<2^{{\omega }_1}$” the existence of a $T_1$ separable locally compact $\left(a\right)$-space with an uncountable closed discrete subset implies the existence of inner models with measurable cardinals. We also use cardinal invariants...

In answer to a question of M. Reed, E. van Douwen and M. Wage [vDW79] constructed an example of a Moore space which had property D but was not pseudonormal. Their construction used the Martin’s Axiom type principle $P\left(c\right)$. We show that there is no such space in the usual Cohen model of the failure of CH.

In this paper we construct a Kelley continuum $X$ such that $X\times [0,1]$ is not semi-Kelley, this answers a question posed by J.J. Charatonik and W.J. Charatonik in A weaker form of the property of Kelley, Topology Proc. 23 (1998), 69–99. In addition, we show that the hyperspace $C\left(X\right)$ is not semi- Kelley. Further we show that small Whitney levels in $C\left(X\right)$ are not semi-Kelley, answering a question posed by A. Illanes in Problemas propuestos para el taller de Teoría de continuos y sus hiperespacios, Queretaro, 2013.