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In mathematics, and particularly general topology, the half-disk topology is an example of a topology given to the set ''X'', given by all points (''x'',''y'') in the plane such that . The set ''X'' can be termed the closed upper half plane. To give the set ''X'' a topology means to say which subsets of ''X'' are "open", and to do so in a way that the following axioms are met: # The union of open sets is an open set. # The finite intersection of open sets is an open set. # The set ''X'' and the empty set ∅ are open sets. == Construction == We consider ''X'' to consist of the open upper half plane ''P'', given by all points (''x'',''y'') in the plane such that ; and the ''x''-axis ''L'', given by all points (''x'',''y'') in the plane such that . Clearly ''X'' is given by the union The open upper half plane ''P'' has a topology given by the Euclidean metric topology.〔 We extend the topology on ''P'' to a topology on by adding some additional open sets. These extra sets are of the form where (''x'',0) is a point on the line ''L'' and ''U'' is an open, with respect to the Euclidean metric topology, neighbourhood of (''x'',''y'') in the plane.〔 == References == 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「Half-disk topology」の詳細全文を読む スポンサード リンク
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