Convention:Hausdorffness assumption

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 Note: Earlier we did make that assumption, so some content is outdated and not synced with the latest assumptions.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.