Attilio Le Donne

Connectifications of metrizable spaces

Abstract We answer a question of Alas, Tkacenko, Tkachuk, and Wilson by constructing a metrizable space with no compact open subsets which cannot be densely embedded in a connected metrizable (or even perfectly normal) space. We also obtain a result that implies that every nowhere locally compact metrizable space can be densely embedded in a connected metrizable space.

Pytkeev spaces and sequential extensions

Abstract In this paper we construct, in response to a question of Malykhin and Tironi, a ZFC example of a perfectly normal Pytkeev space which has no sequential extensions.

Dense embeddings in pathwise connected spaces

Abstract This paper is devoted to the problem of finding those T 1 -spaces (Hausdorff spaces) which are densely embeddable in a pathwise connected T 1 -space (Hausdorff space). In particular, we prove that a countable first countable Hausdorff space (with more than one point) is pathwise connectifiable (i.e., it can be densely embedded in a pathwise connected Hausdorff space) if and only if it has no isolated points. Moreover some examples are given to show that a Hausdorff space which can be densely embedded in a connected Hausdorff space may fail to be pathwise connectifiable.

On locally connected connectifications

Abstract A connected Hausdorff space Y is called a connectification of a space X if X can be densely embedded in Y. This paper is a contribution to the problem of finding those spaces which have a locally connected connectification. The results obtained imply a positive answer to the following questions of Alas, Tkacenko, Tkachuk and Wilson: (i) Does the Sorgenfrey line have a locally connected connectification with countable remainder? (ii) Let X be a countable Hausdorff space without isolated points. Does X have a locally connected connectification?

Partial n-point sets and zero-dimensionality

Abstract In this paper we show that every one-dimensional partial n -point set in higher dimensions contains arcs. This answers a question posed by Dijkstra and van Mill.

Embeddings into normal first countable spaces

Abstract In this paper we construct, in response to a question of Arhangelskii, a zero-dimensional first countable space which cannot be embedded into a normal first countable space.

Weak B-property in Σ-products

Abstract It is shown that each Σ-product of paracompact p-spaces has the weak B -property.