mathematics, a Lindelöf space is a topological spacein which every open coverhas 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 spaceis a Lindelöf space, but not conversely.
open subspaceof 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
Any σ-compact space is Lindelöf.
Properties of strongly Lindelöf spaces
second-countable spaceis a strongly Lindelöf space
Suslin spaceis strongly Lindelöf.
*Strongly Lindelöf spaces are closed under taking countable unions, subspaces, and continuous images.
Radon measureon 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 planeS, which is the product of R under the half-open interval topologywith 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.
open coveringof 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) × −"x", −"x" + 2)
The thing to notice here is that each rectangle "x", "x" + 2) × −"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
uncountablediscrete 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.
The following definition generalises the definitions of compact and Lindelöf: a topological space is κ"-compact" (or κ"-Lindelöf"), where κ is any cardinal, if every open cover has a subcover of cardinality "strictly" less than κ. Compact is then -compact and Lindelöf is then -compact.
The smallest cardinal κ such that a topological space "X" is κ-compact is called"compactness degree" of the space "X". The closely related "Lindelöf degree", l(X), is the smallest cardinal κ such that every open cover of the space X has a subcover of size at most κ. In this notation, X is Lindelöf iff .
axioms of countability
* 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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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
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