Editing Pointed space (section)

{{Short description|Topological space with a distinguished point}}
{{Refimprove|date=November 2009}}

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 <math>x_0,</math> that remains unchanged during subsequent discussion, and is kept track of during all operations.

Maps of pointed spaces ('''based maps''') are [[Continuous (topology)|continuous maps]] preserving basepoints, i.e., a map <math>f</math> between a pointed space <math>X</math> with basepoint <math>x_0</math> and a pointed space <math>Y</math> with basepoint <math>y_0</math> is a based map if it is continuous with respect to the topologies of <math>X</math> and <math>Y</math> and if <math>f\left(x_0\right) = y_0.</math> This is usually denoted
:<math>f : \left(X, x_0\right) \to \left(Y, y_0\right).</math>
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]].