A note on topological properties of non-Hausdorff manifolds
In this short note, the authors introduce the notion of compatible apparition points for a non-Hausdorff manifold. Let \(M\) be a non-Hausdorff manifold and let \(x\), \(y\) be distinct points of \(M\). Then \(x\) and \(y\) are compatible apparition points if there do not exist disjoint open sets \(U\) and \(V\) with \(x \in U\) and \(y \in V\). A set of compatible apparition points is a set such that any pair of distinct points in that set are compatible apparition points. The authors prove some elementary properties of such sets. They show that a set of compatible apparition points in a non-Hausdorff manifold \(M\) is nowhere dense in \(M\). They also show that if \(M\) contains a non-empty set \(S\) of compatible apparition points, then every continuous function from \(M\) to a Hausdorff space \(X\) is constant on \(S\).
scientific article; zbMATH DE number 5627905