Sigmacompactness

Sigmacompactness is a topological property related to the coverability of a topological space by a specific type of cover. A topological space is said to be sigmacompact if it can be written as the countable union of compact subspaces. In other words, if X is a topological space, then X is sigmacompact if there exists a sequence of compact subsets $K_1, K_2, K_3, \ldots$ such that $X = \bigcup_{n=1}^{\infty} K_n$.

This property is a generalization of compactness. Every compact space is trivially sigmacompact, as it is a countable union of itself, and a compact space is compact. The concept of sigmacompactness is particularly useful in certain areas of topology and analysis, especially when dealing with non-compact spaces that still possess some "boundedness" properties inherited from compact sets.

Metrizable spaces that are sigmacompact are precisely the separable spaces. This connection highlights the significance of sigmacompactness within the context of metric spaces. Another important characteristic is that a topological space is sigmacompact if and only if every open cover contains a countable subfamily whose union is the entire space. This latter characterization often proves to be a powerful tool for proving results involving sigmacompactness.