Editing Category of topological spaces (section)

==Limits and colimits==

The category '''Top''' is both [[complete category|complete and cocomplete]], which means that all small [[limit (category theory)|limits and colimit]]s exist in '''Top'''. In fact, the forgetful functor ''U'' : '''Top''' → '''Set''' uniquely lifts both limits and colimits and preserves them as well. Therefore, (co)limits in '''Top''' are given by placing topologies on the corresponding (co)limits in '''Set'''.

Specifically, if ''F'' is a [[diagram (category theory)|diagram]] in '''Top''' and  (''L'', ''φ'' : ''L'' → ''F'') is a limit of ''UF'' in '''Set''', the corresponding limit of ''F'' in '''Top''' is obtained by placing the [[initial topology]] on (''L'', ''φ'' : ''L'' → ''F''). Dually, colimits in '''Top''' are obtained by placing the [[final topology]] on the corresponding colimits in '''Set'''.

Unlike many ''algebraic'' categories, the forgetful functor ''U'' : '''Top''' → '''Set''' does not create or reflect limits since there will typically be non-universal [[cone (category theory)|cones]] in '''Top''' covering universal cones in '''Set'''.

Examples of limits and colimits in '''Top''' include:

*The [[empty set]] (considered as a topological space) is the [[initial object]] of '''Top'''; any [[singleton (mathematics)|singleton]] topological space is a [[terminal object]]. There are thus no [[zero object]]s in '''Top'''.
*The [[product (category theory)|product]] in '''Top''' is given by the [[product topology]] on the [[Cartesian product]]. The [[coproduct (category theory)|coproduct]] is given by the [[disjoint union (topology)|disjoint union]] of topological spaces.
*The [[equaliser (mathematics)#In category theory|equalizer]] of a pair of morphisms is given by placing the [[subspace topology]] on the set-theoretic equalizer. Dually, the [[coequalizer]] is given by placing the [[quotient topology]] on the set-theoretic coequalizer.
*[[Direct limit]]s and [[inverse limit]]s are the set-theoretic limits with the [[final topology]] and [[initial topology]] respectively.
*[[Adjunction space]]s are an example of [[pushout (category theory)|pushouts]] in '''Top'''.