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mildly Hausdorff をテンプレートにして作成
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開始行:
*Definition [#y5bcc40b]
-A topological space (X,τ) is called mildly Hausdorff if the [[δ-closed sets>δ-closed]] [[form a network for its toopology τ>network]].
*Property [#v662a9b4]
-For a topological space (X,τ), the following conditions are equivalent;
++X is mildly Hausdorff.
++For any open set U and any point x ∈ U, there is a [[δ-generalized closed set>δ-generalized closed]] F such that x ∈ F ⊆ U.
++Every open set is η-open.
++Every closed set is the intersection of [[δ-open sets>δ-open]].
-Every mildly Hausdorff [[strongly S-closed]] space is [[locally indiscrete]] and hence [[extremally disconnected]] and [[(semi-)compact]].
-Let &ref(http://www.eaflux.com/imgtex/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 &ref(http://www.eaflux.com/imgtex/imgtex.fcgi?%5bres=100%
++X is mildly Hausdorff.
++Each X_i is mildly Hausdorff.
-Let (X_i, τ_i) be mildly Hausdorff for each i ∈ I. If &ref(http://www.eaflux.com/imgtex/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
-Every (weakly) mildly Hausdorff space is [[I-compact]].
-mildly Hausdorff ⇒ [[R_0]]
*Reference [#re172cd1]
-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.
終了行:
*Definition [#y5bcc40b]
-A topological space (X,τ) is called mildly Hausdorff if the [[δ-closed sets>δ-closed]] [[form a network for its toopology τ>network]].
*Property [#v662a9b4]
-For a topological space (X,τ), the following conditions are equivalent;
++X is mildly Hausdorff.
++For any open set U and any point x ∈ U, there is a [[δ-generalized closed set>δ-generalized closed]] F such that x ∈ F ⊆ U.
++Every open set is η-open.
++Every closed set is the intersection of [[δ-open sets>δ-open]].
-Every mildly Hausdorff [[strongly S-closed]] space is [[locally indiscrete]] and hence [[extremally disconnected]] and [[(semi-)compact]].
-Let &ref(http://www.eaflux.com/imgtex/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 &ref(http://www.eaflux.com/imgtex/imgtex.fcgi?%5bres=100%
++X is mildly Hausdorff.
++Each X_i is mildly Hausdorff.
-Let (X_i, τ_i) be mildly Hausdorff for each i ∈ I. If &ref(http://www.eaflux.com/imgtex/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
-Every (weakly) mildly Hausdorff space is [[I-compact]].
-mildly Hausdorff ⇒ [[R_0]]
*Reference [#re172cd1]
-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.
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