mildly Hausdorff

Definition

• A topological space (X,τ) is called mildly Hausdorff if the δ-closed sets form a network for its toopology τ.

Property

• For a topological space (X,τ), the following conditions are equivalent;
  1. X is mildly Hausdorff.
  2. For any open set U and any point x ∈ U, there is a δ-generalized closed set? F such that x ∈ F ⊆ U.
  3. Every open set is η-open.
  4. Every closed set is the intersection of δ-open sets.
• Every mildly Hausdorff strongly S-closed? space is locally indiscrete? and hence extremally disconnected? and (semi-)compact?.
• Let be a family of Topological spaces. For the topological sum the following conditions are equivalent:
  1. X is mildly Hausdorff.
  2. Each X_i is mildly Hausdorff.
• Let (X_i, τ_i) be mildly Hausdorff for each i ∈ I. If and τ is the product topology on X, then X is mildly Hausdorff.
• Every (weakly) mildly Hausdorff space is I-compact?.

• mildly Hausdorff ⇒ R_0

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.