Connected and/or topological group pd-examples

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 T2 pd-example; (3) there is a T2 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 T3 pd-example, as well as to the existence of an abelian T2 topological group pd-example. However, S≠∅ in itself is not sufficient to imply the existence of a connected T3.5 pd-example. But if there is μ∈S with μ≥c then there is an abelian T2 topological group (hence T3.5) 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.