In mathematics, 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. Lindelöf spaces are named after the FinnishmathematicianErnst Leonard Lindelöf.
Properties of Lindelöf spaces
In general, no implications hold 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 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. Any σ-compact space is Lindelöf.
Properties of strongly Lindelöf spaces
Any second-countable space is a strongly Lindelöf space
Strongly Lindelöf spaces are closed undertaking countable unions, subspaces, and continuous images.
Every Radon measure on a strongly Lindelöf space is moderated.
Product of Lindelöf spaces
The product of Lindelöf spaces is not necessarily Lindelöf. The usual example of this is the Sorgenfrey plane, which is the product of the real line under the half-open interval topology with itself. Open sets in the Sorgenfrey plane are unions of half-open rectangles that include the south and west edges and omit the north and east edges, including the northwest, northeast, and southeast corners. The antidiagonal of is the set of points such that. Consider the open covering of which consists of:
The set of all rectangles, where is on the antidiagonal.
The set of all rectangles, where is on the antidiagonal.
The thing to notice here is that each point on the antidiagonal is contained in exactly one set of the covering, so all these sets are needed. Another way to see that is not Lindelöf is to note that the antidiagonal defines a closed and uncountablediscrete subspace of. This subspace is not Lindelöf, and so the whole space cannot be Lindelöf as well. The product of a Lindelöf space and a compact space is Lindelöf.
Generalisation
The following definition generalises the definitions of compact and Lindelöf: a topological space is -compact, where is any cardinal, if every open cover has a subcover of cardinality strictly less than. Compact is then -compact and Lindelöf is then -compact. The Lindelöf degree, or Lindelöf number, is the smallest cardinal such that every open cover of the space has a subcover of size at most. In this notation, is Lindelöf if. The Lindelöf number as defined above does not distinguish between compact spaces and Lindelöf non compact spaces. Some authors gave the name Lindelöf number to a different notion: the smallest cardinal such that every open cover of the space has a subcover of size strictly less than. In this latter sense the Lindelöf number is the smallest cardinal such that a topological space is -compact. This notion is sometimes also called the compactness degree of the space.
