In mathematics, a pointed space or based space is a topological space with a distinguished point, the basepoint. The distinguished point is 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. Common examples of naturally pointed spaces include spheres (with the north pole as basepoint), closed intervals [0,1] (with 0 as basepoint), and topological groups (with the identity element as basepoint). Maps of pointed spaces (based maps) 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 ( x 0 ) = y 0 . {\displaystyle f\left(x_{0}\right)=y_{0}.} This is usually denoted
f : ( X , x 0 ) → ( Y , y 0 ) . {\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.
Category of pointed spaces The class of all pointed spaces forms a category Top ∙ {\displaystyle \bullet } with basepoint-preserving continuous maps as morphisms. Another way to think about this category is as the comma category, ( { ∙ } ↓ {\displaystyle \{\bullet \}\downarrow } Top) where { ∙ } {\displaystyle \{\bullet \}} is any one point space and Top is the category of topological spaces. (This is also called a coslice category denoted { ∙ } / {\displaystyle \{\bullet \}/} Top.) Objects in this category are continuous maps { ∙ } → X . {\displaystyle \{\bullet \}\to X.} Such maps can be thought of as picking out a basepoint in X . {\displaystyle X.} Morphisms in ( { ∙ } ↓ {\displaystyle \{\bullet \}\downarrow } 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 f {\displaystyle f} preserves basepoints. As a pointed space, { ∙ } {\displaystyle \{\bullet \}} is a zero object in Top { ∙ } {\displaystyle \{\bullet \}} , while it is only a terminal object in Top. There is a forgetful functor Top { ∙ } {\displaystyle \{\bullet \}} → {\displaystyle \to } Top which "forgets" which point is the basepoint. This functor has a left adjoint which assigns to each topological space X {\displaystyle X} the disjoint union of X {\displaystyle X} and a one-point space { ∙ } {\displaystyle \{\bullet \}} whose single element is taken to be the basepoint.
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