# Pointed space

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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 $x_{0},$ 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 $f$ between a pointed space $X$ with basepoint $x_{0}$ and a pointed space $Y$ with basepoint $y_{0}$ is a based map if it is continuous with respect to the topologies of $X$ and $Y$ and if $f\left(x_{0}\right)=y_{0}.$ This is usually denoted

$f:\left(X,x_{0}\right)\to \left(Y,y_{0}\right).$ 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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