Covering compacta by discrete subspaces
Abstract For any space X, denote by dis ( X ) the smallest (infinite) cardinal κ such that κ many discrete subspaces are needed to cover X. It is easy to see that if X is any crowded (i.e. dense-in-itself) compactum then dis ( X ) ⩾ m , where m denotes the additivity of the meager ideal on the reals. It is a natural, and apparently quite difficult, question whether in this inequality m could be replaced by c . Here we show that this can be done if X is also hereditarily normal. Moreover, we prove the following mapping theorem that involves the cardinal function dis ( X ) . If f : X → Y is a continuous surjection of a countably compact T 2 space X onto a perfect T 3 space Y then | { y ∈ Y : f −1 y  is countable } | ⩽ dis ( X ) .
Two improvements on Tkaÿcenko's addition theorem
We prove that (A) if a countably compact space is the union of countably many D subspaces then it is compact; (B) if a compact T2 space is the union of fewer than N(R) = cov(M) left-separated subspaces then it is scattered. Both (A) and (B) improve results of Tkaycenko from 1979; (A) also answers a question that was raised by Arhangel'skiyi and improves a result of Gruenhage.
D-spaces and covering properties
Abstract We study the D -space property and its generalizations, the notions of an aD -space and a weak aD -space in connection with covering properties. A brief survey on D -spaces is presented in Section 1. Among new results, it is proved that if a linearly ordered space is an aD -space, then it is paracompact. The statement further extends the list of equivalences in [Proc. Amer. Math. Soc. 125 (1997) 1237]. We also establish some sufficient conditions for the free topological group of a Tychonoff space to be a D -space. In particular, the free topological group of a semi-stratifiable space is shown to be a D -space, while it need not be semi-stratifiable. A similar result is established for the free topological group of a space with a point-countable base. Some new interesting open problems on D -spaces and on spaces close to them are formulated. In particular, we discuss several such questions in connection with the sum theorems.