“Locally-at” as a Topological Quantifier-Former

Abstract

The language of topology abounds with references to properties that obtain locally (in a certain region) rather than globally (of a space as a whole). Assertions of local truth can often be understood as relativisations of the universal (global) quantifier to neighbourhoods of points. The phenomenon is well illustrated by the notion of the germ of a set. Two sets A and B are said to have the same germ at a point x in a topological space if there is some open set N around x on which A and B agree, i.e. A ∩ N = B ∩ N.

This paper was written while the author held a Nuffield Fellowship and was a visitor to the Mathematical Institute, Oxford.