Point-set topology in ℝn

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In any mathematical study, the first thing to do is specify the type or category of objects to be investigated. In topology, the most general possible objects of study are sets of points with just enough structure to be able to define continuous functions. This, however, leads to a far more abstract study than is our purpose. We will, in this chapter, restrict our attention to subsets of real n-space, ℝn, with all the natural structure to which we are accustomed. This provides a class of objects with structure simple enough to be amenable to study while sufficiently rich in possibilities. Point-set topology is rather like looking at these sets of points under a microscope and identifying characteristics visible in that context. As in the beginning of any study, a great number of definitions, notation, and examples must be absorbed before proceeding. This pattern is rather like that commonly followed in calculus classes, where it is necessary to thoroughly understand limits and local behavior before proceeding to differentiation and integration. By basing everything firmly on the structure of euclidean space, many of the intricacies of general point-set topology can be avoided, but I have tried not to go so far in that direction as to give proper mathematicians the horrors. A number of simple proofs are included in the exercises.KeywordsInterior PointLimit PointTopological PropertyStereographic ProjectionRelative NeighborhoodThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.