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mildly Hausdorff

Last-modified: 2013-08-24 (土) 00:22:59

Definition Edit

Property Edit

  • 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 imgtex.fcgi?%5bres=100%5d%7b%5c%5b%20%28X_i%29_%7bi%5cin%20I%7d%20%5c%5d%7d%25.png be a family of Topological spaces. For the topological sum imgtex.fcgi?%5bres=100%5d%7b%5c%5b%20X=%5ctextstyle%7b%5csum_%7bi%5cin%20I%7d%7dX_i%20%5c%5d%7d%25.png 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 imgtex.fcgi?%5bres=100%5d%7b%5c%5b%20X=%5ctextstyle%7b%5cprod_%7bi%5cin%20I%7d%7dX_i%20%5c%5d%7d%25.png 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 Edit

  • 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.