Convention:Hausdorffness assumption

From Topospaces
Jump to: navigation, search

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

We do not assume normal spaces or regular spaces to satisfy the T1 or Hausdorff conditions. Note: Earlier we did make that assumption, so some content is outdated and not synced with the latest assumptions.

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.