Pointed space

In mathematics, a pointed 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 formula_1 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 formula_2 between a pointed space formula_3 with basepoint formula_4 and a pointed space formula_5 with basepoint formula_6 is a based map if it is continuous with respect to the topologies of formula_3 and formula_5 and if formula_9 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.
Category of pointed spaces.
The class of all pointed spaces forms a category Topformula_11 with basepoint preserving continuous maps as morphisms. Another way to think about this category is as the comma category, (formula_12 Top) where formula_13 is any one point space and Top is the category of topological spaces. (This is also called a coslice category denoted formula_14Top.) Objects in this category are continuous maps formula_15 Such morphisms can be thought of as picking out a basepoint in formula_16 Morphisms in formula_12 Top) are morphisms in Top for which the following diagram commutes:
It is easy to see that commutativity of the diagram is equivalent to the condition that formula_2 preserves basepoints.
As a pointed space, formula_13 is a zero object in Topformula_13, while it is only a terminal object in Top.
There is a forgetful functor Topformula_13 formula_22 Top which "forgets" which point is the basepoint. This functor has a left adjoint which assigns to each topological space formula_3 the disjoint union of formula_3 and a one-point space formula_13 whose single element is taken to be the basepoint.