Editing Pointed space (section)

== Category of pointed spaces ==

The [[Class (set theory)|class]] of all pointed spaces forms a [[Category (mathematics)|category]] '''Top'''<sub><math>\bull</math></sub> with basepoint preserving continuous maps as [[morphism]]s. Another way to think about this category is as the [[comma category]], (<math>\{ \bull \} \downarrow </math> '''Top''') where <math>\{ \bull \}</math> is any one point space and '''Top''' is the [[category of topological spaces]]. (This is also called a [[coslice category]] denoted <math>\{ \bull \} /</math>'''Top'''.) Objects in this category are continuous maps <math>\{ \bull \} \to X.</math> Such maps can be thought of as picking out a basepoint in <math>X.</math> Morphisms in (<math>\{ \bull \} \downarrow </math> '''Top''') are morphisms in '''Top''' for which the following diagram [[Commutative diagram|commutes]]:

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[[Image:PointedSpace-01.png]]
</div>

It is easy to see that commutativity of the diagram is equivalent to the condition that <math>f</math> preserves basepoints.

As a pointed space, <math>\{ \bull \}</math> is a [[zero object]] in '''Top'''<sub><math>\{ \bull \}</math></sub>, while it is only a [[terminal object]] in '''Top'''.

There is a [[forgetful functor]] '''Top'''<sub><math>\{ \bull \}</math></sub> <math>\to</math> '''Top''' which "forgets" which point is the basepoint. This functor has a [[Adjoint functor|left adjoint]] which assigns to each topological space <math>X</math> the [[disjoint union]] of <math>X</math> and a one-point space <math>\{ \bull \}</math> whose single element is taken to be the basepoint.