A topology T for a space X gives a rough notion of proximity (eventually relevant for sequences and analysis) and also induces a system of directed arrows on the power set. Continuous functions are an easily definable class of maps special for that topology from or to X and those in turn often translate to morphisms of object on that space. The definitions is actually so rudimentary that it gives rise to dualities all over the place (see Stone duality, Gelfand representation theorems, models for non-classical logics, and stronger notions of topologies (Grothendieck topoi) merge with an even broader class of relevant mathematical objects) It's also a tool to classify sets with other structure (or let's say spaces).
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