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# Lindelöf space

In mathematics, a Lindelöf space is a topological space in which every open cover has a countable subcover. A Lindelöf space 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 subset is 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. Also, 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 &sigma;-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 S, which is the product of 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.

Consider the open covering of S which consists of:

# The set of all points ("x", "y") with "x" < "y"
# The set of all points ("x", "y") with "x" + 1 > "y"
# For each real "x", the half-open rectangle "x", "x" + 2) &times; −"x", −"x" + 2)

The thing to notice here is that each rectangle "x", "x" + 2) &times; −"x", −"x" + 2) covers exactly one of the points on the line "x" = −"y". None of the points on this line is included in any of the other sets in the cover, so there is no proper subcover of this cover, which therefore contains no countable subcover.Another way to see that S is not Lindelöf is to note that the line "x" = −"y" 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 smallest cardinal &kappa; such that a topological space "X" is &kappa;-compact is called"compactness degree" of the space "X". The closely related "Lindelöf degree", 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 iff $l\left(X\right) = aleph_0$.

ee also

*axioms of countability
*Lindelöf's lemma

References

* Michael Gemignani, "Elementary Topology" (ISBN 0-486-66522-4) (see especially section 7.2)
* | year=1995
*cite book | author=I. Juhász | title=Cardinal functions in topology - ten years later | publisher=Math. Centre Tracts, Amsterdam | year=1980 | id=ISBN 90-6196-196-3

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### Look at other dictionaries:

• Lindelöf space — /lin deuh lof , lawf , luef /, Math. a topological space having the property that every cover consisting of open sets has a subcover consisting of a countable number of subsets. [after Ernst Lindelöf (1870 1946), Finnish topologist] * * * …   Universalium

• Lindelöf space — /lin deuh lof , lawf , luef /, Math. a topological space having the property that every cover consisting of open sets has a subcover consisting of a countable number of subsets. [after Ernst Lindelöf (1870 1946), Finnish topologist] …   Useful english dictionary

• Lindelöf space — noun A topological space such that any open cover of the space has a countable subcover …   Wiktionary

• space — 1. noun /speɪs/ a) The intervening contents of a volume. If it be only a Single Letter or two that drops, he thruſts the end of his Bodkin between every Letter of that Word, till he comes to a Space: and then perhaps by forcing thoſe Letters… …   Wiktionary

• Lindelöf's lemma — In mathematics, Lindelöf s lemma is a simple but useful lemma in topology on the real line, named for the Finnish mathematician Ernst Leonard Lindelöf.tatement of the lemmaLet the real line have its usual Borel topology. Then every open subset of …   Wikipedia

• Ernst Leonard Lindelöf — Ernst Leonard Lindelöf, (7 March 1870 ndash;4 June 1946) was a Finnish topologist after whom Lindelöf space is named; he was the son of Leonard Lorenz Lindelöf and brother of the philologist Uno Lorenz Lindelöf.Lindelöf studied at the University… …   Wikipedia

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• Ernst Leonard Lindelöf —  Ne doit pas être confondu avec le producteur américain de télévision Damon Lindelof. Ernst Leonard Lindelöf (7 mars 1870 – 4 juin 1946), est un mathématicien finlandais qui travailla principalement en analyse complexe et en théorie des… …   Wikipédia en Français

• Compact space — Compactness redirects here. For the concept in first order logic, see compactness theorem. In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness… …   Wikipedia

• Paracompact space — In mathematics, a paracompact space is a topological space in which every open cover admits a locally finite open refinement. Paracompact spaces are sometimes also required to be Hausdorff. Paracompact spaces were introduced by Dieudonné (1944).… …   Wikipedia