weakly Hausdorff

Definition

• A topological space (X,τ) is called weakly Hausdorff if each singleton is δ-closed.

Property

• A topological space (X,τ) is weakly Hausdorff if and only if every subset is η-open.
• For a topological space (X,τ) the following conditions are equivalent;
  1. X is weakly Hausdorff.
  2. X is almost weakly Hausdorff and mildly Hausdorff.
  3. X is almost weakly Hausdorff and weakly mildly Hausdorff.

• Every weakly Hausdorff S-closed? space is extremally disconnected? and Urysohn.

Reference

• Dontchev, J.; Popvassilev, S.; Stavrova, D., On the η-expansion topology for the co-semi-regularization and mildly Hausdorff spaces., (English summary), Acta Math. Hungar. 80 (1998), no. 1-2, 9-19.