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In mathematics, in the field of topology, a topological space is said to be a door space if every subset is either open or closed (or both).〔Kelley, ch.2, Exercise C, p. 76.〕 The term comes from the introductory topology mnemonic that "a subset is not like a door: it can be open, closed, both, or neither". Here are some easy facts about door spaces: * A Hausdorff door space has at most one accumulation point. * In a Hausdorff door space if x is not an accumulation point then is open. To prove the first assertion, let X be a Hausdorff door space, and let x ≠ y be distinct points. Since X is Hausdorff there are open neighborhoods U and V of x and y respectively such that U ∩ V = ∅. Suppose y is an accumulation point. Then U \ ∪ is closed, since if it were open, then we could say that = (U \ ∪ ) ∩ V is open, contradicting that y is an accumulation point. So we conclude that as U \ ∪ is closed, X \ (U \ ∪ ) is open and hence = U ∩ () is open, implying that x is not an accumulation point. == Notes == 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「Door space」の詳細全文を読む スポンサード リンク
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