| Category Of Topological Spaces |
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| CATEGORIES ABOUT CATEGORY OF TOPOLOGICAL SPACES | |
| category-theoretic categories | |
| topological spaces | |
| general topology | |
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N.B. Some authors use the name Top for the category with Topological Manifold s 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 Set s with additional structure (i.e. topologies) and its morphisms are Function s preserving this structure. There is a natural Forgetful Functor U to the Category Of Sets which assigns to each topological space the underlying set and to each continuous map the underlying Function . LIMITS AND COLIMITS The category Top is both complete and cocomplete, which means that all small Limits and Colimit s 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:
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