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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Kuratowski closure axioms ： ウィキペディア英語版 Kuratowski closure axioms In topology and related branches of mathematics, the Kuratowski closure axioms are a set of axioms that can be used to define a topological structure on a set. They are equivalent to the more commonly used open set definition. They were first introduced by Kazimierz Kuratowski.〔 〕 A similar set of axioms can be used to define a topological structure using only the dual notion of interior operator. == Definition == Let $X$ be a set and $\backslash mathcal(X)$ its power set. A Kuratowski Closure Operator is an assignment $\backslash operatorname:\backslash mathcal(X)\; \backslash to\; \backslash mathcal(X)$ with the following properties:〔 has a fifth (optional) axiom stating that singleton sets are their own closures. He refers to topological spaces which satisfy all five axioms as T1 spaces in contrast to the more general spaces which only satisfy the four listed axioms.〕 # $\backslash operatorname(\backslash varnothing)\; =\; \backslash varnothing$ (Preservation of Nullary Union) # $A\; \backslash subseteq\; \backslash operatorname(A)\; \backslash textA\; \backslash subseteq\; X$ (Extensivity) # $\backslash operatorname(A\; \backslash cup\; B)\; =\; \backslash operatorname(A)\; \backslash cup\; \backslash operatorname(B)\; \backslash textA,B\; \backslash subseteq\; X$ (Preservation of Binary Union) # $\backslash operatorname(\backslash operatorname(A))\; =\; \backslash operatorname(A)\; \backslash textA\; \backslash subseteq\; X$ (Idempotence) If the last axiom, idempotence, is omitted, then the axioms define a preclosure operator. A consequence of the third axiom is: $A\; \backslash subseteq\; B\; \backslash Rightarrow\; \backslash operatorname(A)\; \backslash subseteq\; \backslash operatorname(B)$ (Preservation of Inclusion). The four Kuratowski closure axioms can be replaced by a single condition, namely, :$A\; \backslash cup\; \backslash operatorname(A)\; \backslash cup\; \backslash operatorname(\backslash operatorname(B))\; =\; \backslash operatorname(A\; \backslash cup\; B)\; \backslash setminus\; \backslash operatorname(\backslash varnothing)\; \backslash textA,\; B\; \backslash subseteq\; X.$ 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Kuratowski closure axioms」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.