Lindelöf space

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 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. 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
Any Suslin space is strongly Lindelöf.
Strongly Lindelöf spaces are closed under taking 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 $\mathbb {S}$ , which is the product of the real line $\mathbb {R}$ 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 $\mathbb {S}$ is the set of points $(x,y)$ such that $x+y=0$ .

Consider the open covering of $\mathbb {S}$ which consists of:

The set of all rectangles $(-\infty ,x)\times (-\infty ,y)$ , where $(x,y)$ is on the antidiagonal.
The set of all rectangles $[x,+\infty )\times [y,+\infty )$ , 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 $S$ is not Lindelöf is to note that the antidiagonal defines a closed and uncountable discrete subspace of $S$ . This subspace is not Lindelöf, and so the whole space cannot be Lindelöf as well (as closed subspaces of Lindelöf spaces are also Lindelöf).

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 $\kappa$ -compact (or $\kappa$ -Lindelöf), where $\kappa$ is any cardinal, if every open cover has a subcover of cardinality strictly less than $\kappa$ . Compact is then $\aleph _{0}$ -compact and Lindelöf is then $\aleph _{1}$ -compact.

The Lindelöf degree, or Lindelöf number $l(X)$ , is the smallest cardinal $\kappa$ such that every open cover of the space $X$ has a subcover of size at most $\kappa$ . In this notation, $X$ is Lindelöf if $l(X)=\aleph _{0}$ . 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 $\kappa$ such that every open cover of the space $X$ has a subcover of size strictly less than $\kappa$ .^[1] In this latter (and less used) sense the Lindelöf number is the smallest cardinal $\kappa$ such that a topological space $X$ is $\kappa$ -compact. This notion is sometimes also called the compactness degree of the space $X$ .^[2]

Notes

^ Mary Ellen Rudin, Lectures on set theoretic topology, Conference Board of the Mathematical Sciences, American Mathematical Society, 1975, p. 4, retrievable on Google Books [1]
^ Hušek, Miroslav (1969), "The class of k-compact spaces is simple", Mathematische Zeitschrift, 110: 123–126, doi:10.1007/BF01124977, MR 0244947.

References

Michael Gemignani, Elementary Topology (ISBN 0-486-66522-4) (see especially section 7.2)
Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1995) [1978]. Counterexamples in Topology (Dover reprint of 1978 ed.). Berlin, New York: Springer-Verlag. ISBN 978-0-486-68735-3. MR 0507446.
I. Juhász (1980). Cardinal functions in topology - ten years later. Math. Centre Tracts, Amsterdam. ISBN 90-6196-196-3.
Munkres, James. Topology, 2nd ed.
http://arxiv.org/abs/1301.5340 Generalized Lob's Theorem.Strong Reflection Principles and Large Cardinal Axioms.Consistency Results in Topology

[1] Mary Ellen Rudin, Lectures on set theoretic topology, Conference Board of the Mathematical Sciences, American Mathematical Society, 1975, p. 4, retrievable on Google Books [1]

[2] Hušek, Miroslav (1969), "The class of k-compact spaces is simple", Mathematische Zeitschrift, 110: 123–126, doi:10.1007/BF01124977, MR 0244947.

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