Basic Object: Topological spaces
Basic Map: Continuous functions
Historically, much of point set topology was developed to understand the correct definitions for such notions as continuity and dimension. By now, though, these definitions permeate mathematics, frequently in areas seemingly far removed from the traditional topological space Rn. Unfortunately, it is not at first apparent that these more abstract definitions are at all useful; there needs to be an initial investment in learning the basic terms. In the first section, these basic definitions are given. In the next section, these definitions are applied to the topological space Rn, where all is much more down to earth. Then we look at metric spaces. The last section applies these definitions to the Zariski topology of a commutative ring, which, while natural in algebraic geometry and algebraic number theory, is not at all similar to the topology of Rn.
Much of point set topology consists in developing a convenient language to talk about when various points in a space are near to one another and about the notion of continuity. The key is that the same definitions can be applied to many disparate branches of math.
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