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In mathematics, the category of topological spaces, often denoted Top, is the category whose objects are topological spaces and whose morphisms are continuous maps. This is a category because the composition of two continuous maps is again continuous. The study of Top and of properties of topological spaces using the techniques of category theory is known as categorical topology.
N.B. Some authors use the name Top for the category with topological manifolds as objects and continuous maps as morphisms.
Top is a concrete category
Like many categories, the category Top is a concrete category, meaning its objects are sets with additional structure (i.e. topologies) and its morphisms are functions preserving this structure. There is a natural forgetful functor
- U : Top → Set
Limits and colimits
The category Top is both complete and cocomplete, which means that all small limits and colimits exist in Top.
The forgetful functor U : Top → Set has a left adjoint which equips a given set with the discrete topology and a right adjoint which equips a given set with the trivial topology. This implies that the functor U is both limit-preserving and colimit-preserving, i.e. limits in Top are given by placing topologies on the corresponding limits 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.
Other properties
- The monomorphisms in Top are the injective continuous maps, the epimorphisms are the surjective continuous maps, and the isomorphisms are the homeomorphisms.
- The extremal monomorphisms are (essentially) the subspace embeddings. Every extremal monomorphism is regular.
- The extremal epimorphisms are (essentially) the quotient maps. Every extremal epimorphism is regular.
- There are no zero morphisms in Top, and in particular the category is not preadditive.
- Top is not cartesian closed (and therefore also not a topos) since it does not have exponential objects for all spaces.
Relationships to other categories
- The category of pointed topological spaces Top• is a coslice category over Top.
- The homotopy category hTop has topological spaces for objects and homotopy equivalence classes of continuous maps for morphisms. This is a quotient category of Top. One can likewise form the pointed homotopy category hTop•.
- Top contains the important category Haus of topological spaces with the Hausdorff property as a full subcategory. It should be noted that the added structure of this subcategory allows for more epimorphisms: in fact, the epimorphisms in this subcategory are precisely those morphisms with dense images in their codomains, so that epimorphisms need not be surjective.
References
- Adámek, Jiří, Herrlich, Horst, & Strecker, George E.; (1990). Abstract and Concrete Categories (4.2MB PDF). Originally publ. John Wiley & Sons. ISBN 0-471-60922-6. (now free on-line edition).
"Category of topological spaces"