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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 x 0 , {\displaystyle x_{0},} that remains unchanged during subsequent discussion, and is kept track of during all operations.
Maps of pointed spaces are continuous maps preserving basepoints, i.e., a map f {\displaystyle f} between a pointed space X {\displaystyle X} with basepoint x 0 {\displaystyle x_{0}} and a pointed space Y {\displaystyle Y} with basepoint y 0 {\displaystyle y_{0}} is a based map if it is continuous with respect to the topologies of X {\displaystyle X} and Y {\displaystyle Y} and if f = y 0 . {\displaystyle f\left=y_{0}.} 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.