Parthasarathy, shift-compactness and infinite combinatorics

Abstract

Parthasarathy’s heritage has a hidden component arising from a variant of his concept of shift-compactness which yields quick proofs of fundamental theorems reviewed here. We demonstrate the closeness of the variant notion to his original one as arising in the tacit treatment of all possible convergent centrings. We also include a very short proof of the Effros Mapping Theorem – a non-linear version of the Open Mapping Theorem. This is deduced from a shift-compactness theorem. Both these can be given a constructive form by implementing a constructive improvement to a theorem on the separation of points and closed nowhere dense sets.