| 摘 要： | 1. This note aims to introduce the notion of the categories with topologies and to discuss topologies generated by functors. A topological semigroup is a category with topologies, in addition, Propositions 10 and 11 will show the generated topologies compatible with adjoint equivalences and bijective half-functors (which are generalized inverse operations in an n-preadditive category, see 1]). Thus, we can believe that there are natural connections between topological and algebraic structures in category structure. In this direction, we shall propose several papers and this note will be an introduction of them.
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