A Lindelöf space is a topological space in which every open cover has a countable subcover.
A Lindelöf space is a topological space in which every open cover has a countable subcover. The Lindelöf property is a weakening of the more commonly used notion of compactness, which requires the existence of a finite subcover.
A strongly Lindelöf space is a topological space such that every open subspace is Lindelöf. Such spaces are also known as hereditarily Lindelöf spaces, because all subspaces of such a space are Lindelöf.
In general, no implications hold (in either direction) between the Lindelöf property and other compactness properties, such as paracompactness. But by the Morita theorem, every regular Lindelöf space is paracompact.
Any second-countable space is a (strongly) Lindelöf space, but not conversely. However, the matter is simpler for metric spaces. A metric space is Lindelöf if and only if it is separable, and if and only if it is second-countable.
An open subspace of a Lindelöf space is not necessarily Lindelöf. In particular in a Lindelöf space, every open subspace is Lindelöf if and only if every subspace is Lindelöf. However, a closed subspace must be Lindelöf.
Being Lindelöf is preserved by continuous maps. However, it is not necessarily preserved by products, not even by finite products.
A Lindelöf space is compact if and only if it is countably compact, i.e. every countable open cover has a finite subcover.
Any $\sigma$-compact space is Lindelöf. A topological space is said to be $\sigma$-compact if it is the union of countably many compact subspaces.
Source: Wikipedia
