|
In mathematics, more specifically point-set topology, a Moore space is a developable regular Hausdorff space. Equivalently, a topological space ''X'' is a Moore space if the following conditions hold: * Any two distinct points can be separated by neighbourhoods, and any closed set and any point in its complement can be separated by neighbourhoods. (''X'' is a regular Hausdorff space.) * There is a countable collection of open covers of ''X'', such that for any closed set ''C'' and any point ''p'' in its complement there exists a cover in the collection such that every neighbourhood of ''p'' in the cover is disjoint from ''C''. (''X'' is a developable space.) Moore spaces are generally interesting in mathematics because they may be applied to prove interesting metrization theorems. The concept of a Moore space was formulated by R. L. Moore in the earlier part of the 20th century. ==Examples and properties== #Every metrizable space, ''X'', is a Moore space. If is the open cover of ''X'' (indexed by ''x'' in ''X'') by all balls of radius 1/''n'', then the collection of all such open covers as ''n'' varies over the positive integers is a development of ''X''. Since all metrizable spaces are normal, all metric spaces are Moore spaces. #Moore spaces are a lot like regular spaces and different from normal spaces in the sense that every subspace of a Moore space is also a Moore space. #The image of a Moore space under an injective, continuous open map is always a Moore space. Note also that the image of a regular space under an injective, continuous open map is always regular. #Both examples 2 and 3 suggest that Moore spaces are a lot similar to regular spaces. #Neither the Sorgenfrey line nor the Sorgenfrey plane are Moore spaces because they are normal and not second countable. #The Moore plane (also known as the Niemytski space) is an example of a non-metrizable Moore space. #Every metacompact, separable, normal Moore space is metrizable. This theorem is known as Traylor’s theorem. #Every locally compact, locally connected space, normal Moore space is metrizable. This theorem was proved by Reed and Zenor. #If , then every separable normal Moore space is metrizable. This theorem is known as Jones’ theorem. 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「Moore space (topology)」の詳細全文を読む スポンサード リンク
|