I was wondering if there are some necessary and sufficient conditions for the quotient space to be Haussdorf. I have been trying a little for a while, but I only got very restricti …

A Boolean algebra may, or may not, be complete (i.e, any set of elements has a sup and an inf) or atomic (i.e., every element is a sup of some set of atoms). Boolean Algebras tha …

The question arose while comparing the notions of compactness, countable compactness, local compactness, and "Lindelofness" in Hausdorff spaces. It is straightforward to show that …

I found it very hard to find literature about smooth manifolds that are not required to be Hausdorff. In particular I'm interested in their local properties: 1.) Are the $r$-th or …

This question is about the space of all topologies on a fixed set X. We may order the topologies by refinement, so that τ ≤ σ just in case every τ open set is open …

In principle a sequence in a non-Hausdorff space can converge to two points simultaneously. Can anyone give me an explicit example of the above? Or tell me any method of generati …

A Hausdorff space $(X,\tau)$ is said to be minimal Hausdorff if for each topology $\tau' \subseteq \tau$ with $\tau' \neq \tau$ the space $(X,\tau')$ is not Hausdorff. Every compa …

In his book "Riemannian Geometry" do Carmo cites the Hopf-Rinow theorem in chapter 7. (theorem 2.8). One of the equivalences there deals with the cover of the manifold using nested …

does every infinite hausdorff space contains a countable infinite discrete subspace?