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In topology, a branch of mathematics, the ends of a topological space are, roughly speaking, the connected components of the “ideal boundary” of the space. That is, each end represents a topologically distinct way to move to infinity within the space. Adding a point at each end yields a compactification of the original space, known as the end compactification. ==Definition== Let ''X'' be a topological space, and suppose that : ''K''1 ⊂ ''K''2 ⊂ ''K''3 ⊂ · · · is an ascending sequence of compact subsets of ''X'' whose interiors cover ''X''. Then ''X'' has one end for every sequence : ''U''1 ⊃ ''U''2 ⊃ ''U''3 ⊃ · · ·, where each ''U''''n'' is a connected component of ''X'' \ ''K''''n''. The number of ends does not depend on the specific sequence of compact sets; there is a natural bijection between the sets of ends associated with any two such sequences. Using this definition, a neighborhood of an end is an open set ''V'' such that ''V'' ⊃ ''U''''n'' for some ''n''. Such neighborhoods represent the neighborhoods of the corresponding point at infinity in the end compactification (this “compactification” isn’t always compact; the topological space ''X'' has to be connected and locally connected). The definition of ends given above applies only to spaces ''X'' that possess an exhaustion by compact sets (that is, ''X'' must be hemicompact). However, it can be generalized as follows: let ''X'' be any topological space, and consider the direct system of compact subsets of ''X'' and inclusion maps. There is a corresponding inverse system , where ''π''0(''Y'') denotes the set of connected components of a space ''Y'', and each inclusion map ''Y'' → ''Z'' induces a function ''π''0(''Y'') → ''π''0(''Z''). Then set of ends of ''X'' is defined to be the inverse limit of this inverse system. Under this definition, the set of ends is a functor from the category of topological spaces where morphisms are only ''proper'' continuous maps to the category of sets. Explicitly, if φ : X → Y is a proper map and ''x''=(''x''''K'')K is an end of ''X'' (ie each element ''x''''K'' in the family is a connected component of ''X'' ∖ ''K'' and they are compatible with maps induced by inclusions) then φ(x) is the family where ranges over compact subsets of ''Y'' and φ * is the map induced by φ from to . Properness of φ is used to ensure that each φ⁻¹(''K'') is compact in ''X''. The original definition above represents the special case where the direct system of compact subsets has a cofinal sequence. 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「End (topology)」の詳細全文を読む スポンサード リンク
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