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In mathematics, an adjunction space (or attaching space) is a common construction in topology where one topological space is attached or "glued" onto another. Specifically, let ''X'' and ''Y'' be topological spaces with ''A'' a subspace of ''Y''. Let ''f'' : ''A'' → ''X'' be a continuous map (called the attaching map). One forms the adjunction space ''X'' ∪''f'' ''Y'' by taking the disjoint union of ''X'' and ''Y'' and identifying ''x'' with ''f''(''x'') for all ''x'' in ''A''. Schematically, : Sometimes, the adjunction is written as . Intuitively, we think of ''Y'' as being glued onto ''X'' via the map ''f''. As a set, ''X'' ∪''f'' ''Y'' consists of the disjoint union of ''X'' and (''Y'' − ''A''). The topology, however, is specified by the quotient construction. In the case where ''A'' is a closed subspace of ''Y'' one can show that the map ''X'' → ''X'' ∪''f'' ''Y'' is a closed embedding and (''Y'' − ''A'') → ''X'' ∪''f'' ''Y'' is an open embedding. ==Examples== *A common example of an adjunction space is given when ''Y'' is a closed ''n''-ball (or ''cell'') and ''A'' is the boundary of the ball, the (''n''−1)-sphere. Inductively attaching cells along their spherical boundaries to this space results in an example of a CW complex. *Adjunction spaces are also used to define connected sums of manifolds. Here, one first removes open balls from ''X'' and ''Y'' before attaching the boundaries of the removed balls along an attaching map. *If ''A'' is a space with one point then the adjunction is the wedge sum of ''X'' and ''Y''. *If ''X'' is a space with one point then the adjunction is the quotient ''Y''/''A''. 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「Adjunction space」の詳細全文を読む スポンサード リンク
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