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.
Contents
- Properties of Lindelf spaces
- Properties of strongly Lindelf spaces
- Product of Lindelf spaces
- Generalisation
- References
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 for the Finnish mathematician Ernst Leonard Lindelöf.
Properties of Lindelöf spaces
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 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. However, a closed subspace must be Lindelöf.
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
Product of Lindelöf spaces
The product of Lindelöf spaces is not necessarily Lindelöf. The usual example of this is the Sorgenfrey plane
Consider the open covering of
- The set of all rectangles
( − ∞ , x ) × ( − ∞ , y ) , where( x , y ) is on the antidiagonal. - The set of all rectangles
[ x , + ∞ ) × [ y , + ∞ ) , where( x , y ) 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
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
The Lindelöf degree, or Lindelöf number