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Encyclopedia of Separation Axioms Wiki*
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T_0 をテンプレートにして作成
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開始行:
*Definition [#cc4edb6a]
-A topological space (X,τ) is T_0 if every pair of distinct points is [[topologically distinguishable]].
*Property [#q3c8946e]
-A topological space (X,τ) is a T_0-space iff one of the following conditions holds:
++∀x in X, the [[derived set]] {x}' is a union of closed sets. [2]
++∀x in X, cl({x}) ∩ ker(x) = {x}. [2]
++∀x in X, {x}' = D{x} where D{x} is the [[essential derived set]] of x. [2]
++∀x, y in X, y ∈ {x}' implies &ref(http://www.eaflux.com/imgtex/imgtex.fcgi?%5bres=100%5d%7b%5c%5b%20x%20%5cnotin%20%5c%7by%5c%7d'%20%5c%5d%7d%25.png);. [2]
++∀x, y in X, y ∈ {x}' implies &ref(http://www.eaflux.com/imgtex/imgtex.fcgi?%5bres=100%5d%7b%5c%5b%20%5c%7by%5c%7d'%5csubsetneq%5crm%7bD%7d%5c%7bx%5c%7d%20%5c%5d%7d%25.png);. [2]
++∀x, y in X, y ∈ {x}' implies y ∈ D{x}. [2]
++∀x, y in X, &ref(http://www.eaflux.com/imgtex/imgtex.fcgi?%5bres=100%5d%7b%5c%5b%20x%20%5cneq%20y%20%5cmbox%7b%20implies%20%7d%20%5crm%7bcl%7d%28%5c%7bx%5c%7d%29%5ccap%20%5crm%7bker%7d%28x%29%5cneq%5crm%7bcl%7d%28%5c%7by%5c%7d%29%5ccap%20
-T_0 ⇔ [[λ-T_1]]. [4]
-T_0 ⇒ [[λ-T_0]]. [4]
-T_0 = [[T_R]] + [[R*_0]]. [3]
*Reference [#d18118b2]
+??
+Guia, Josep, ''Axioms weaker than R0.'', (Serbo-Croatian summary), Mat. Vesnik 36 (1984), no. 3, 195–205.
+Guia, Josep, ''Essentially T_D and essentially T_UD spaces.'', Bull. Math. Soc. Sci. Math. R. S. Roumanie (N.S.) 32(80) (1988), no. 3, 227-233.
+Caldas Miguel, Jafari Saeid, Navalagi Govindappa, ''More on λ-closed sets in topological spaces.'',Rev. Colombiana Mat. 41 (2007), no. 2, 355-369.
終了行:
*Definition [#cc4edb6a]
-A topological space (X,τ) is T_0 if every pair of distinct points is [[topologically distinguishable]].
*Property [#q3c8946e]
-A topological space (X,τ) is a T_0-space iff one of the following conditions holds:
++∀x in X, the [[derived set]] {x}' is a union of closed sets. [2]
++∀x in X, cl({x}) ∩ ker(x) = {x}. [2]
++∀x in X, {x}' = D{x} where D{x} is the [[essential derived set]] of x. [2]
++∀x, y in X, y ∈ {x}' implies &ref(http://www.eaflux.com/imgtex/imgtex.fcgi?%5bres=100%5d%7b%5c%5b%20x%20%5cnotin%20%5c%7by%5c%7d'%20%5c%5d%7d%25.png);. [2]
++∀x, y in X, y ∈ {x}' implies &ref(http://www.eaflux.com/imgtex/imgtex.fcgi?%5bres=100%5d%7b%5c%5b%20%5c%7by%5c%7d'%5csubsetneq%5crm%7bD%7d%5c%7bx%5c%7d%20%5c%5d%7d%25.png);. [2]
++∀x, y in X, y ∈ {x}' implies y ∈ D{x}. [2]
++∀x, y in X, &ref(http://www.eaflux.com/imgtex/imgtex.fcgi?%5bres=100%5d%7b%5c%5b%20x%20%5cneq%20y%20%5cmbox%7b%20implies%20%7d%20%5crm%7bcl%7d%28%5c%7bx%5c%7d%29%5ccap%20%5crm%7bker%7d%28x%29%5cneq%5crm%7bcl%7d%28%5c%7by%5c%7d%29%5ccap%20
-T_0 ⇔ [[λ-T_1]]. [4]
-T_0 ⇒ [[λ-T_0]]. [4]
-T_0 = [[T_R]] + [[R*_0]]. [3]
*Reference [#d18118b2]
+??
+Guia, Josep, ''Axioms weaker than R0.'', (Serbo-Croatian summary), Mat. Vesnik 36 (1984), no. 3, 195–205.
+Guia, Josep, ''Essentially T_D and essentially T_UD spaces.'', Bull. Math. Soc. Sci. Math. R. S. Roumanie (N.S.) 32(80) (1988), no. 3, 227-233.
+Caldas Miguel, Jafari Saeid, Navalagi Govindappa, ''More on λ-closed sets in topological spaces.'',Rev. Colombiana Mat. 41 (2007), no. 2, 355-369.
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