Convention:Hausdorffness assumption

This article is about a convention that is followed in this wiki. The aim is that every page on the wiki follows this convention unless explicitly stated otherwise on the page; however, in practice, this may not have been implemented

For separation axioms

In the Topospaces wiki, we follow the convention that Hausdorffness is assumed for all higher separation axioms. In particular, normal spaces and regular spaces are assumed to have the Hausdorffness condition. This convention is compatible with a number of general-purpose textbooks in point-set topology and algebraic topology, including those by Munkres, Singer-Thorpe, Hatcher, among others.

People working in point-set topology research sometimes do not assume the Hausdorffness condition for normal spaces, so please keep this in mind.

For compactness axioms

For compactness-type properties, we do not assume Hausdorffness. Thus, a compact space need not be Hausdorff. This convention is again in line with a number of general-purpose textbooks in point-set topology, but is not in line with terminology in algebraic geometry, which often follows the Bourbaki convention of calling a compact non-Hausdorff space quasi-compact.

For manifolds

We assume manifolds to be Hausdorff. This convention is again in line with most treatises on point-set topology, algebraic topology, and differential topology. The non-Hausdorff versions are called locally Euclidean spaces here.