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In mathematics, the particular point topology (or included point topology) is a topology where sets are considered open if they are empty or contain a particular, arbitrarily chosen, point of the topological space. Formally, let ''X'' be any set and ''p'' ∈ ''X''. The collection :''T'' = of subsets of ''X'' is then the particular point topology on ''X''. There are a variety of cases which are individually named: * If ''X'' = we call ''X'' the Sierpiński space. This case is somewhat special and is handled separately. * If ''X'' is finite (with at least 3 points) we call the topology on ''X'' the finite particular point topology. * If ''X'' is countably infinite we call the topology on ''X'' the countable particular point topology. * If ''X'' is uncountable we call the topology on ''X'' the uncountable particular point topology. A generalization of the particular point topology is the closed extension topology. In the case when ''X'' \ has the discrete topology, the closed extension topology is the same as the particular point topology. This topology is used to provide interesting examples and counterexamples. ==Properties== ; Closed sets have empty interior : Given an open set every is a limit point of A. So the closure of any open set other than is . No closed set other than contains p so the interior of every closed set other than is . 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「particular point topology」の詳細全文を読む スポンサード リンク
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