A remark on proper partitions of unity

TL;DR

This paper introduces a new concept of proper partitions of unity within the framework of exterior spaces, generalizing classical notions through a categorical approach and an extension of Alexandroff compactification, with applications to fiberwise proper homotopy.

Contribution

It develops an analogue of numerable coverings in the proper and exterior setting using categorical methods and compactification generalizations.

Findings

Defined a proper partition of unity in exterior spaces.

Extended the Alexandroff compactification to this context.

Applied the concept to fiberwise proper homotopy equivalences.

Abstract

In this paper we introduce, by means of the category of exterior spaces and using a process that generalizes the Alexandroff compactification, an analogue notion of numerable covering of a space in the proper and exterior setting. An application is given for fibrewise proper homotopy equivalences.