- #1

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Topological space is defined as a set ##X## and a collection ##T## of its subsets that satisfies some general properties (which I will not list here, but I will assume that the reader knows them). But such a very general definition of a topological space does not seem to capture the intuitive idea that topology is about distinguishing objects that cannot continuously be transformed into each other (e.g. coffee cup and torus are topologically the same). What's the motivation for such a general definition of a topological space?

Even more confusing is that the sets in ##T## are

*called*open sets. Why are they called so, given that some of them can be closed intervals?