# Connected and/or topological group pd-examples

## TL;DR

This paper explores the existence of topological spaces with small pinning down numbers relative to density, constructing examples with properties like connectivity and group structure, linked to set-theoretic conditions.

## Contribution

It introduces constructions transforming pd-examples into connected, topological group, and vector space examples, relating their existence to set-theoretic assumptions about singular cardinals.

## Key findings

- Connected and locally connected pd-examples exist under certain set-theoretic conditions.
- Existence of abelian topological group pd-examples is equivalent to non-emptiness of a class of singular cardinals.
- Conditions on singular cardinals imply the existence of arcwise connected and locally convex pd-examples.

## Abstract

The pinning down number $pd(X)$ of a topological space $X$ is the smallest cardinal $\kappa$ such that for every neighborhood assignment $\mathcal{U}$ on $X$ there is a set of size $\kappa$ that meets every member of $\mathcal{U}$. Clearly, $pd(X) \le d(X)$ and we call $X$ a pd-example if $pd(X) < d(X)$. We denote by $\mathbf{S}$ the class of all singular cardinals that are not strong limit. It was proved in a paper of Juh\'asz,Soukup and Szentmikl\'ossy (arXiv:1506.00206}) that TFAE: (1) $\mathbf{S} \ne \emptyset$; (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 $\mathbf{S} \ne \emptyset$ 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, $\mathbf{S} \ne \emptyset$ in itself is not sufficient to imply the existence of a connected $T_{3.5}$ pd-example. But if there is $\mu \in \mathbf{S}$ with $\mu \ge \mathfrak{c}$ 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 $\,\mathbf{S} \setminus \mathfrak{c} \ne \emptyset\,$ even implies that there is a locally convex topological vector space pd-example.