Compact Metric Spaces

A metric space (M, d) is said to be compact if it is both complete and totally bounded. As you might imagine, a compact space is the best of all possible worlds.

Examples 8.1

1. (a) A subset K of ℝ is compact if and only if K is closed and bounded. This fact is usually referred to as the Heine–Borel theorem. Hence, a closed bounded interval [a, b] is compact. Also, the Cantor set Δ is compact. The interval (0, 1), on the other hand, is not compact.

2. (b) A subset K of ℝn is compact if and only if K is closed and bounded. (Why?)

3. (c) It is important that we not confuse the first two examples with the general case. Recall that the set {en:n ≥ 1} is closed and bounded in ℓ∞ but not totally bounded – hence not compact. Taking this a step further, notice that the closed ball {x: ∥x∥∞ ≤ 1} in ℓ∞ is not compact, whereas any closed ball in ℝn is compact.

4. (d) A subset of a discrete space is compact if and only if it is finite. (Why?)

Just as with completeness and total boundedness, we will want to give several equivalent characterizations of compactness. In particular, since neither completeness nor total boundedness is preserved by homeomorphisms, our newest definition does not appear to be describing a topological property.