Sigmacompact

Sigmacompact (also written sigma‑compact) is a topological property defined for a space that can be expressed as a countable union of compact subspaces. More precisely, a topological space X is sigmacompact if there exists a sequence of compact subsets {Kₙ} such that X = ⋃ₙ Kₙ. The notation σ corresponds to the Greek letter sigma, denoting a countable union. This property is weaker than compactness but stronger than being Lindelöf in many common contexts.

Introduced in the early 20th century by Henri Lebesgue and later studied by many authors, sigma‑compactness plays a significant role in general topology, descriptive set theory, and functional analysis. It ensures that many arguments relying on compactness can be extended to larger spaces by using a countable decomposition. For example, every locally compact second‑countable space is sigma‑compact, since a countable base allows the construction of a countable exhaustion by compact neighborhoods.

Key properties include: sigma‑compactness is preserved by continuous surjections; it is hereditary with respect to closed subspaces; and the product of a sigma‑compact space with a compact space remains sigma‑compact. However, the product of two sigma‑compact spaces need not be sigma‑compact, as demonstrated by the Sorgenfrey line, which is sigma‑compact but its self‑product is not.

Common examples of sigma‑compact spaces are the Euclidean spaces ℝⁿ, all σ‑compact Lie groups, and complete separable metric spaces (Polish spaces). Non‑examples include the uncountable discrete space, which cannot be written as a countable union of compact subsets because each finite subset is compact and an uncountable space would require uncountably many such subsets.