Exercise 1062Topology › Metric and normed spacesL1Added 3 Nov 2026

Compactness in metric spaces

#compactness#open cover#sequential

Statement

State the equivalence between compactness by open covers and sequential compactness in metric spaces.

Hint

In metric spaces, compact $\Leftrightarrow$ sequentially compact.

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Solution

Theorem. For a metric space $X$ : compact (every open cover has a finite subcover) $\iff$ every sequence has a convergent subsequence (sequential compactness).

Key steps. Sequential compactness implies the space is totally bounded; a totally bounded complete space is compact.

Conclusion. Both compactness notions coincide in metric spaces.

Compactness in metric spaces

Statement

Solution

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