Functional Analysis Lecture Notes: Compact Sets and Finite-dimensional Spaces

Definition 1.1 (Compact and Totally Bounded Sets). Let X be a metric space, and let E ⊆ X be given. (a) We say that E is compact if every open cover of E contains a finite subcover. That is, E is compact if whenever {Uα}α∈I is a collection of open sets whose union contains E, then there exist finitely many α1, . . . , αN such that E ⊆ Uα1 ∪ · · · ∪ UαN . (b) We say that E is sequentially compact if every sequence {fn}n∈N of points of E contains a convergent subsequence {fnk}k∈N whose limit belongs to E. (c) We say that E is totally bounded if for every ε > 0 there exist finitely many points f1, . . . , fN ∈ E such that E ⊆ N