Metric Spaces and Uniform Structures

The general notion of topology does not allow to compare neighborhoods of different points. Such a comparison is quite natural in various geometric contexts. The general setting for such a comparison is that of a uniform structure. The most common and natural way for a uniform structure to appear is via a metric, which was already mentioned on several occasions in Chapter 1, so we will postpone discussing the general notion of union structure to Section 3.11 until after detailed exposition of metric spaces. Another important example of uniform structures is that of topological groups, see Section 3.12 below in this chapter. Also, as in turns out, a Hausdorff compact space carries a natural uniform structure, which in the separable case can be recovered from any metric generating the topology. Metric spaces and topological groups are the notions central for foundations of analysis.