Spaces from pieces. IV (Q1346383)

In categories that are topological over sets with respect to injections, the property is considered whether there exists a structure \(s\) on \(X\) such that its restrictions to some subsets \(X_ i \subset X\) induce given structures \(s_ i\) on \(X_ i\). In the first part, some general conditions are found for categories to have the described property. The remaining parts apply a general situation to about 32 structures, like generalized closure spaces, proximities and uniformities.