D-8 - The Lindelöf Property

Abstract

A topological space is said to be a Lindelöf space or is said to have the Lindelöf property if every open cover of X has a countable sub cover. The Lindelöf property was introduced by Alexandroff and Urysohn in 1929, with the term ëLindelöfí referring back to Lindelöf's result that any family of open subsets of Euclidean space has a countable sub-family with the same union. Clearly, a space is compact if it is both Lindelöf and countably compact, though weaker properties—for example, pseudo compactness—imply compactness in the presence of the Lindelöf property. It is an important result that regular Lindelöf spaces are paracompact, from which it follows that they are collection wise normal. Conversely, every paracompact space with a dense Lindelöf subspace is Lindelöf (in particular, every separable paracompact space is Lindelöf) and every locally compact, paracompact space is a disjoint sum of open Lindelöf subspaces. A related result is that any locally finite family of subsets of a Lindelöf space is countable. A space is compact if and only if (iff) every infinite subset has a complete accumulation point, iff every increasing open cover has an infinite sub cover, and a space is countably compact if every countably infinite subset has a complete accumulation point.