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| CATEGORIES ABOUT POINTED SPACE | |
| topology | |
| homotopy theory | |
| category-theoretic categories | |
f 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 Morphism s. 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. Note that 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. OPERATIONS ON POINTED SPACES
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