The Latest Web Hosting, VPS & Dedicated Server Reviews for 2016.Q: Reference request: Classifying cofibrations in $\textbf{Top}^1$ Let $\textbf{Top}^1$ be the category of topological spaces which are totally disconnected (closed and bounded subsets are compact). There is a functor $\textbf{Top} \to \textbf{Top}^1$ which takes a space $X$ to the space $X^+$ consisting of the disjoint union of $X$ and one point, topologized so that the point is closed and the open sets of $X$ and their images under the inclusion $X \to X^+$ are compact. The functor is called the “compactification functor”. This functor has a left adjoint which takes a space $Y$ and collapses it to the one point. This operation is called “making $Y$ compact”. Consider now the case where $Y$ is not discrete. Then there are two options: either $Y$ is totally disconnected and the map $Y \to \{pt\}$ is open, or there is one point in $Y$ which is not closed (and so not compact). It is the second case that we want to be interested in for the following reasons: If $\{pt\}$ is compact (in the usual topology) and $Y$ is totally disconnected, then $Y$ is a point. Hence the only cofibrations of $\textbf{Top}^1$ are those between points and thus are just the trivial cofibrations. The point of the following question is to find the cofibrations of $\textbf{Top}^1$ in the first case and which they are, i.e. show that they are precisely those inclusions $Y \to \{pt\}$ of a compact space into a point. In fact this question is a part of an extensive geometric attempt to understand the notion of weak homotopy equivalence. I have done some research but came up with only these two references which deal with the compactification functor in the case of a discrete target and none in the case of a totally disconnected target. So I am left with two questions: Are the classifications in 1. and 2. correct? Are there other references on this topic?