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Sequentially Compact Set

Definition

Let (X,dX)(X,d_{X}) be a metric space. A subset EE of XX is termed sequentially compact if for every sequence (xi)i=1∞(x_i)_{i=1}^\infty with each xi∈E,x_i \in E\text{,} there exists a convergent subsequence (xij)j=1∞(x_{i_j})_{j=1}^\infty that converges to a point x∈E.x \in E\text{.}

Interpretation

A set is sequentially compact if every sequence drawn from it has a subsequence that converges to a point still inside the set. Sequences in the set cannot escape, neither to infinity nor to a missing point at its edge.

Two discs. In the closed disc a sequence has a subsequence converging to a point inside the set. In the open disc the points creep towards a rim point outside the set, so no subsequence converges in it.

In the closed disc, the red subsequence converges to a point of EE. In the open disc, the illustrated sequence approaches a missing rim point, and every subsequence has that same limit outside EE, so the open disc is not sequentially compact.

Two requirements are packed into the definition: a convergent subsequence must exist, and its limit must lie in EE. The first fails for unbounded sets, as xi=ix_i=i in R\mathbb{R} shows; the second fails for sets that are not closed, as for the open disc. The whole sequence need not converge: xi=(−1)ix_i=(-1)^i in E={−1,1}E=\{-1,1\} does not, but its even-indexed terms do.

In Rn\mathbb{R}^n, the Bolzano-Weierstrass theorem shows that the sequentially compact sets are exactly the closed and bounded ones. In every metric space, sequential compactness is equivalent to compactness, so either description can be used. Sequences are often more convenient in proofs, such as showing that a continuous function attains its maximum, while the definition with open covers also makes sense where no metric is available.

A common slip is to think of sequential compactness as being "closed under limits". That describes closed sets, and it is not enough: R\mathbb{R} is closed, but xi=ix_i=i has no convergent subsequence.

Key Results

  • In any metric space, a set is compact if and only if it is sequentially compact. This gives two equivalent ways to think about compactness: open covers or convergent subsequences.

Examples

  • The closed unit ball {x∈Rn ⁣:∥x∥≤1}\{\boldsymbol{x}\in \mathbb{R}^n\colon \|\boldsymbol{x}\|\leq 1\} is sequentially compact. This fails in infinite dimensions.
  • Any finite set in any metric space is sequentially compact: some point is taken by infinitely many terms of a sequence in it, which gives a constant subsequence.

Prerequisites

Sources

Keywords: sequential compactness Symbols: element of, infinity, real numbers

Last updated: 20 October 2026 03:02