Definitions and examples

A topological space is a more basic concept than a metric space. Its building blocks are open sets, as suggested by the work for real numbers along the lines of that in Section 1.6.

The abstract idea of a metric space provides a useful and quite visual example of a topological space. Through much of this chapter, we will relate our work to corresponding ideas in metric spaces. In previous chapters, we have spent some time on closed sets and compact sets. These were defined specifically in the context of metric spaces, and each definition made use of the notion of a convergent sequence. The same terms will be used again in this chapter, but they will be redefined in the more general context of topological spaces. To distinguish the different approaches, we will be careful in this chapter to refer to the earlier notions as sequentially closed sets and sequentially compact sets.

So a set is sequentially closed if convergent sequences in the metric space that belong to the set have their limits in the set, and a set is sequentially compact if every sequence in the set has a convergent subsequence. These are the old definitions; new ones will come soon. It will turn out, and these are two of the important results of this chapter, that the old definitions and the new definitions coincide in metric spaces.

The term ‘topology’ refers to the work of this chapter in general, but is also used in the technical sense given by the following definition.