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One of the most useful concepts for analysis that arise from topology and metric spaces is the concept of compactness; recall that a space ${X}$ is compact if every open cover of ${X}$ has a finite subcover, or equivalently if any collection of closed sets with the finite intersection property (i.e. every finite subcollection of these sets has non-empty intersection) has non-empty intersection. In these notes, we explore how compactness interacts with other key topological concepts: the Hausdorff property, bases and sub-bases, product spaces, and equicontinuity, in particular establishing the useful Tychonoff and Arzelá-Ascoli theorems that give criteria for compactness (or precompactness).

Exercise 1 (Basic properties of compact sets)

• Show that any finite set is compact.
• Show that any finite union of compact subsets of a topological space is still compact.
• Show that any image of a compact space under a continuous map is still compact.

Show that these three statements continue to hold if “compact” is replaced by “sequentially compact”.