Glueing spaces without identifying points

Abstract

In this paper we develop the theory of Artin-Wraith glueings for topological spaces. It means that we are glueing two topological spaces to each other without identifying points on them. These constructions are far from unique and this theory is used to deal with all of them.As an application, we show that some categories of compactifications of coarse spaces that agree with the coarse structures are invariant under coarse equivalences. As a consequence, if X and Y are some well behaved metric spaces that are coarse equivalent, then they have the same space of ends (generalizing the well known fact that works on quasi-isometric proper geodesic metric spaces). As another application, we show that for every compact metrizable space Y, there exists only one, up to homeomorphisms, compactification of the Cantor set minus one point such that the remainder is homeomorphic to Y.