Pointed space

Topological space with a distinguished point / From Wikipedia, the free encyclopedia

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In mathematics, a pointed space or based space is a topological space with a distinguished point, the basepoint. The distinguished point is just simply one particular point, picked out from the space, and given a name, such as that remains unchanged during subsequent discussion, and is kept track of during all operations.

Maps of pointed spaces (based maps) are continuous maps preserving basepoints, i.e., a map between a pointed space with basepoint and a pointed space with basepoint is a based map if it is continuous with respect to the topologies of and and if This is usually denoted

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.

Pointed spaces are often taken as a special case of the relative topology, where the subset is a single point. Thus, much of homotopy theory is usually developed on pointed spaces, and then moved to relative topologies in algebraic topology.

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