Category of Topological Spaces - Limits and Colimits

Limits and Colimits

The category Top is both complete and cocomplete, which means that all small limits and colimits 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 in Top and (L, φ) is a limit of UF in Set, the corresponding limit of F in Top is obtained by placing the initial topology on (L, φ). 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 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 topological space is a terminal object. There are thus no zero objects in Top.
• The product in Top is given by the product topology on the Cartesian product. The coproduct is given by the disjoint union of topological spaces.
• The 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 limits and inverse limits are the set-theoretic limits with the final topology and initial topology respectively.
• Adjunction spaces are an example of pushouts in Top.