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

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

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.