ABSTRACT
Mathematics would certainly have not come into existence if one had known from the beginning that there was in nature no exactly straight line, no actual circle, no absolute magnitude.
Friedrich Nietzsche
To someone unfamiliar with the subject, the topology of spaces such as R, Rk , or C can be a difficult concept to describe. Our desire in this introductory note is only to motivate our study of certain topological topics as they relate to analysis. The chapter itself will serve to make these ideas more precise. Also, it is worth emphasizing that many topological results applicable to more general spaces (that is, to spaces other than R, Rk , and C) belong more properly to the subject of topology itself, not analysis, and so we will not pursue them. With these goals in mind, our motivation for the study of topology stems from the following fact: It will be useful in our work to characterize the points of a space in certain ways. In particular, if A is any subset of one of the spaces R, Rk , or C, we will characterize the points of the space relative to A. That is, we will define a collection of properties that describe in some way how the points of A are configured relative to each other, and to the points of the space outside of A. We will collectively refer to the study of such properties as point-set topology. It is both significant and convenient that within the spaces R, Rk , and C, these properties can be described in relation to the distances between points. It is just as significant and convenient that each of these spaces is a normed space, and that within each of them we can specify the distances between points using the norm on each space. This shared trait will allow us to efficiently characterize the topological features of these spaces in terms that are common to all of them. For this reason, we will use the symbol X throughout our discussion to represent any one of the spaces R, Rk , or Cwhen the result to which we refer applies to each of them. If a result is unique to one of the spaces R, Rk , or C or does not apply to one of them, we will make explicit mention of that fact.
