|
In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. ==Formal definition== Let ''X'' be a topological space. Most commonly ''X'' is called ''locally compact'', if every point of ''X'' has a compact neighbourhood. There are other common definitions: They are all equivalent if ''X'' is a Hausdorff space (or preregular). But they are not equivalent in general: :1. every point of ''X'' has a compact neighbourhood. :2. every point of ''X'' has a closed compact neighbourhood. :2′. every point of ''X'' has a relatively compact neighbourhood. :2″. every point of ''X'' has a local base of relatively compact neighbourhoods. :3. every point of ''X'' has a local base of compact neighbourhoods. :3′. for every point ''x'' of ''X'', every neighbourhood of ''x'' contains a compact neighbourhood of ''x''. :4. ''X'' is Hausdorff and satisfies any (all) of the previous conditions. Logical relations among the conditions: *Conditions (2), (2′), (2″) are equivalent. *Conditions (3), (3′) are equivalent. *Neither of conditions (2), (3) implies the other. *Each condition implies (1). *Compactness implies conditions (1) and (2), but not (3). Condition (1) is probably the most commonly used definition, since it is the least restrictive and the others are equivalent to it when ''X'' is Hausdorff. This equivalence is a consequence of the facts that compact subsets of Hausdorff spaces are closed, and closed subsets of compact spaces are compact. Condition (4) is used, for example, in Bourbaki. In almost all applications, locally compact spaces are indeed also Hausdorff. These locally compact Hausdorff (LCH) spaces are thus the spaces that this article is primarily concerned with. 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「Locally compact space」の詳細全文を読む スポンサード リンク
|