On characterizing certain classses of first countable spaces by open mappings

On Radon measures on first-countable spaces

It is shown that every Radon measure on a first-countable Hausdorff space is separable provided ω1 is a precaliber of every measurable algebra. As the latter is implied by MA(ω1), the result answers a problem due to D. H. Fremlin. Answering the problem posed by D. H. Fremlin ([4], 32R(c)), we show in this note that, assuming (∗) ω1 is a precaliber of every measurable Boolean algebra, every Rado...

Relative Ranks of Lipschitz Mappings on Countable Discrete Metric Spaces

Let X be a countable discrete metric space and let XX denote the family of all functions on X. In this article, we consider the problem of finding the least cardinality of a subset A of XX such that every element of XX is a finite composition of elements of A and Lipschitz functions on X. It follows from a classical theorem of Sierpiński that such an A either has size at most 2 or is uncountabl...

On nowhere first-countable compact spaces with countable π-weight

The minimum weight of a nowhere first-countable compact space of countable π-weight is shown to be κB, the least cardinal κ for which the real line R can be covered by κ many nowhere dense sets.

A Characterization of Certain Quasi-open Mappings

CH and first countable , countably compact spaces ✩

We show that it is consistent with the Continuum Hypothesis that first countable, countably compact spaces with no uncountable free sequences are compact. As a consequence, we get that CH does not imply the existence of a perfectly normal, countably compact, non-compact space, answering a question of Nyikos (Question 287 in the numbering of van Mill and Reed, Open Problems in Topology, Elsevier...