kompaksekuens

Kompaksekuens is a term found in some Indonesian-language texts on topology to describe a sequence whose values lie inside a compact subset of a topological space. The exact usage can vary by source, and the term is not universally standardized. In many contexts, it is related to the notion of relative compactness.

Definition. Let X be a topological space and (x_n) a sequence in X. The sequence is called kompaksekuens if there exists a compact subset K ⊆ X such that the entire range of the sequence is contained in K; equivalently, {x_n : n ∈ N} ⊆ K. In metric spaces this is often stated as the closure of the set {x_n} being compact.

Key properties. If a sequence is kompaksekuens in a metric space, then it has at least one convergent subsequence by the Bolzano–Weierstrass theorem, with the limit lying in the compact set K. Therefore, kompaksekuens sequences enjoy subsequential convergence. If the ambient space X itself is compact, every sequence in X is kompaksekuens. By contrast, having a convergent subsequence does not by itself guarantee that the whole sequence lies in a single compact subset, so it does not imply kompaksekuens in general.

Relation to related concepts. The idea aligns with relative compactness: a subset is relatively compact when its closure is compact, and a sequence is often described as relatively compact if its range lies in such a subset. In standard terminology, the same idea is discussed via compactness and sequential compactness without necessarily using the exact label kompaksekuens.

Examples. In Euclidean space R^n, any bounded sequence has a compact closure, hence is kompaksekuens. In a compact space, every sequence is kompaksekuens.

Etymology. The term combines "kompak" (compact) with "sekuens" (sequence). See also compactness, sequential compactness, Bolzano–Weierstrass theorem.