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In mathematics, a pointed space is a topological space ''X'' with a distinguished basepoint ''x''0 in ''X''. Maps of pointed spaces (based maps) are continuous maps preserving basepoints, i.e. a continuous map ''f'' : ''X'' → ''Y'' such that ''f''(''x''0) = ''y''0. This is usually denoted :''f'' : (''X'', ''x''0) → (''Y'', ''y''0). Pointed spaces are important in algebraic topology, particularly in homotopy theory, where many constructions, such as the fundamental group, depend on a choice of basepoint. The pointed set concept is less important; it is anyway the case of a pointed discrete space. ==Category of pointed spaces== The class of all pointed spaces forms a category Top• with basepoint preserving continuous maps as morphisms. Another way to think about this category is as the comma category, ( ↓ Top) where is any one point space and Top is the category of topological spaces. (This is also called a coslice category denoted /Top.) Objects in this category are continuous maps → ''X''. Such morphisms can be thought of as picking out a basepoint in ''X''. Morphisms in ( ↓ Top) are morphisms in Top for which the following diagram commutes: It is easy to see that commutativity of the diagram is equivalent to the condition that ''f'' preserves basepoints. As a pointed space is a zero object in Top• while it is only a terminal object in Top. There is a forgetful functor Top• → Top which "forgets" which point is the basepoint. This functor has a left adjoint which assigns to each topological space ''X'' the disjoint union of ''X'' and a one point space whose single element is taken to be the basepoint. 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「Pointed space」の詳細全文を読む スポンサード リンク
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