Connected and/or topological group pd-examples

Abstract

The pinning down number pd(X) of a topological space X is the smallest cardinal κ such that for every neighborhood assignment U on X there is a set of size κ that meets every member of U . Clearly, pd(X) ≤ d(X) and we call X a pd-example if pd(X) < d(X). We denote by S the class of all singular cardinals that are not strong limit. It was proved in [6] that TFAE:(1) S = ∅;(2) there is a 0-dimensional T 2 pd-example;(3) there is a T 2 pd-example. The aim of this paper is to produce pd-examples with further interesting topological properties like connectivity or being a topological group by presenting several constructions that transform given pd-examples into ones with these additional properties.We show that S = ∅ is also equivalent to the existence of a connected and locally connected T 3 pd-example, as well as to the existence of an abelian T 2 topological group pd-example.However, S = ∅ in itself is not sufficient to imply the existence of a connected T 3.5 pd-example. But if there is µ ∈ S with µ ≥ c then there is an abelian T 2 topological group (hence T 3.5 ) pd-example which is also arcwise connected and locally arcwise connected. Finally, the same assumption S \ c = ∅ even implies that there is a locally convex topological vector space pd-example.