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Encyclopedia of Separation Axioms Wiki*
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R_0 をテンプレートにして作成
これらのキーワードがハイライトされています:
開始行:
*Definition [#s1f8b41a]
-A topological space X is a R_0 space if any two [[topologically distinguishable]] points in X can be [[separated]]
*Property [#t5dbb85d]
-A topological space (X, τ) is R_0 if it satisfies one of the following equivalent conditions:
++&ref(http://www.eaflux.com/imgtex/imgtex.fcgi?%5bres=100%5d%7b%5c%5b%20x%5cin%20U%5cin%20%5ctau%20%5c%5d%7d%25.png); implies &ref(http://www.eaflux.com/imgtex/imgtex.fcgi?%5bres=100%5d%7b%5c%5b%20%5cmathrm%7bcl%7d%5c%7bx%5c%7d%5csubset%20
++cl({x})=∩{U | U is an open neighbourhood of x} for each x in X.
++cl({x})=cl({y}) or cl({x}) ⋂ cl({y})= &ref(http://www.eaflux.com/imgtex/imgtex.fcgi?%5bres=100%5d%7b%5c%5b%20%5cemptyset%20%5c%5d%7d%25.png); for x, y in X.
++The [[specialization preoreder]] on X is [[symmetric]].
++The fixed ultrafilter at x convereges only to the points that x is topologically indistinguishable from.
++The [[Kolmogorov quotient]] of X ( which identifies [[topologically indistinguishable]] points) is T_1
++Every open set is the union of closed sets.
-A topological space (X,τ) is R_0 iff for any point x in X, cl({x}) ⊆ ker(x).
-An R_0 space is a [[w-R_0>weakly R_0]] space. The converse does not hold.
-R_0 ⇒ [[C_0]]. [4]
-R_0 ⇒ [[R_T]]. [2]
-R_0 ⇒ [[R_YS]]. [3]
-R_0 ⇒ [[R_D]]. [3]
-R_0 ⇒ [[R*_D]]. [4]
-R_0 + [[normal]] ⇒ regular. [3]
*Reference [#o2756157]
+忘れた。
+Tong, Jing Cheng, ''On the separation axiom R0.'' (Serbo-Croatian summary), Glas. Mat. Ser. III 18(38) (1983), no. 1, 149–152.
+Misra, D. N.; Dube, K. K., ''Some axioms weaker than the R0-axiom.'', (Serbo-Croatian summary), Glasnik Mat. Ser. III 8(28) (1973), 145-148.
+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.
終了行:
*Definition [#s1f8b41a]
-A topological space X is a R_0 space if any two [[topologically distinguishable]] points in X can be [[separated]]
*Property [#t5dbb85d]
-A topological space (X, τ) is R_0 if it satisfies one of the following equivalent conditions:
++&ref(http://www.eaflux.com/imgtex/imgtex.fcgi?%5bres=100%5d%7b%5c%5b%20x%5cin%20U%5cin%20%5ctau%20%5c%5d%7d%25.png); implies &ref(http://www.eaflux.com/imgtex/imgtex.fcgi?%5bres=100%5d%7b%5c%5b%20%5cmathrm%7bcl%7d%5c%7bx%5c%7d%5csubset%20
++cl({x})=∩{U | U is an open neighbourhood of x} for each x in X.
++cl({x})=cl({y}) or cl({x}) ⋂ cl({y})= &ref(http://www.eaflux.com/imgtex/imgtex.fcgi?%5bres=100%5d%7b%5c%5b%20%5cemptyset%20%5c%5d%7d%25.png); for x, y in X.
++The [[specialization preoreder]] on X is [[symmetric]].
++The fixed ultrafilter at x convereges only to the points that x is topologically indistinguishable from.
++The [[Kolmogorov quotient]] of X ( which identifies [[topologically indistinguishable]] points) is T_1
++Every open set is the union of closed sets.
-A topological space (X,τ) is R_0 iff for any point x in X, cl({x}) ⊆ ker(x).
-An R_0 space is a [[w-R_0>weakly R_0]] space. The converse does not hold.
-R_0 ⇒ [[C_0]]. [4]
-R_0 ⇒ [[R_T]]. [2]
-R_0 ⇒ [[R_YS]]. [3]
-R_0 ⇒ [[R_D]]. [3]
-R_0 ⇒ [[R*_D]]. [4]
-R_0 + [[normal]] ⇒ regular. [3]
*Reference [#o2756157]
+忘れた。
+Tong, Jing Cheng, ''On the separation axiom R0.'' (Serbo-Croatian summary), Glas. Mat. Ser. III 18(38) (1983), no. 1, 149–152.
+Misra, D. N.; Dube, K. K., ''Some axioms weaker than the R0-axiom.'', (Serbo-Croatian summary), Glasnik Mat. Ser. III 8(28) (1973), 145-148.
+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.
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