# 2016 seminar talk: Neighborhoods, Compacta and the Tukey Order

Talk held by Paul Gartside (University of Pittsburgh, Pennsylvania, USA) at the KGRC seminar on 2016-05-13.

### Abstract

The neighborhoods of a point or set in a topological space form a filter. The compact subsets of a space form a (pre-)filter. Both are directed sets. In this talk we investigate the properties of these directed sets, and the connections between them. The main tool is the Tukey order. We will focus on separable metrizable spaces, and topological objects derived from them, such as function spaces and free topological groups. An important role will be played by chain conditions, in particular calibre $(\omega_1,\omega)$. We will then turn this around and use topological techniques to construct interesting examples of directed sets with and without certain chain conditions.