Compact Extensions

Abstract

Compactness, in general, is not a hereditary property. The role of compactness in topology is so essential that it is natural to raise the question of the possibility of representing topological spaces as subspaces of compact spaces. The most general and principally the most important question concerning the possibility of embedding a topological space into a compact space can easily be answered in a positive way. Every topological space X can be “extended” to a compact space by adding to it a single point ξ and defining X ∪ {ξ} as the only neighborhood of ξ. Under this definition all open subsets of X are open in X ∪{ξ}. Let us assume that X ≠ Ø. The compact space X ∪ {ξ} containing X as a dense open subspace has one essential deficiency. It does not even satisfy the T₁ separation axiom since none of the sets {x}, where x ∈ X, is closed in X ∪ {ξ}.