Exercise · ZW-SEQUENTIAL-COMPACTNESS-METRIC-SPACES-010 · ★★★ Advanced

Let \(X\) be infinite with the discrete distance (\(d=1\) between distinct points). Is it sequentially compact?

Choose one option.

Show answer

B. No: a sequence of distinct points has no convergent subsequence

Explanation

With the discrete distance a sequence converges only if it is eventually constant. A sequence of distinct points has no constant subsequence, so none converges. Still, \(X\) is bounded, closed and complete.

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Review the concept: Sequential compactness

A metric space \(X\) is sequentially compact if every sequence \((x_n)\) in \(X\) has a subsequence \((x_{n_k})\) converging to a point \(x\in X\).

\[\forall (x_n)\subseteq X\ \ \exists\, n_1

More practice problems: sequential compactness

ZW-SEQUENTIAL-COMPACTNESS-METRIC-SPACES-001 Which of these spaces, with the usual distance, is sequentially compact?
ZW-SEQUENTIAL-COMPACTNESS-METRIC-SPACES-002 Why is ℝ not sequentially compact?
ZW-SEQUENTIAL-COMPACTNESS-METRIC-SPACES-003 To prove that (0,1) is not sequentially compact, which sequence works?
ZW-SEQUENTIAL-COMPACTNESS-METRIC-SPACES-004 In X=(1,2] with the usual distance, does the sequence x_n=1+tfrac1n have a subsequence converging i…
ZW-SEQUENTIAL-COMPACTNESS-METRIC-SPACES-005 Which of these subsets of ℝ^2, with the Euclidean distance, is sequentially compact?
ZW-SEQUENTIAL-COMPACTNESS-METRIC-SPACES-006 Let x_n=cos \!big(tfracnπ2big) in [-1,1]. Which values are limits of some subsequence?
ZW-SEQUENTIAL-COMPACTNESS-METRIC-SPACES-007 Let x_n=(-1)^n (n)/(n+1) in [-1,1]. To which values do its convergent subsequences converge?
ZW-SEQUENTIAL-COMPACTNESS-METRIC-SPACES-008 For a metric space X, which statement is true?