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| In [[mathematics]], a '''Lindelöf space''' is a [[topological space]] in which every [[open cover]] has a [[countable set|countable]] subcover. The Lindelöf property is a weakening of the more commonly used notion of ''[[compact space|compactness]]'', which requires the existence of a ''finite'' subcover.
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| 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 [[Subspace topology|subspaces]] of such a space are Lindelöf.
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| Lindelöf spaces are named for the [[Finland|Finnish]] [[mathematician]] [[Ernst Leonard Lindelöf]].
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| == Properties of Lindelöf spaces ==
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| In general, no implications hold (in either direction) between the Lindelöf property and other compactness properties, such as [[paracompact space|paracompactness]]. But by the Morita theorem, every [[regular space|regular]] Lindelöf space is paracompact.
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| Any [[second-countable space]] is a Lindelöf space, but not conversely. However, the matter is simpler for [[metric space]]s. A metric space is Lindelöf if and only if it is [[separable space|separable]], and if and only if it is [[second-countable space|second-countable]].
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| An [[open subspace]] of a Lindelöf space is not necessarily Lindelöf. However, a closed subspace must be Lindelöf.
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| Lindelöf is preserved by [[continuous function (topology)|continuous maps]]. However, it is not necessarily preserved by products, not even by finite products.
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| A Lindelöf space is compact if and only if it is [[countably compact]].
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| Any [[σ-compact space]] is Lindelöf.
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| == Properties of strongly Lindelöf spaces ==
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| * Any [[second-countable space]] is a strongly Lindelöf space
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| * Any [[Suslin space]] is strongly Lindelöf.
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| * Strongly Lindelöf spaces are closed under taking countable unions, subspaces, and continuous images.
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| * Every [[Radon measure]] on a strongly Lindelöf space is moderated.
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| == Product of Lindelöf spaces ==
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| The [[product space|product]] of Lindelöf spaces is not necessarily Lindelöf. The usual example of this is the [[Sorgenfrey plane]] <math>\mathbb{S}</math>, which is the product of the [[real line]] <math>\mathbb{R}</math> under the [[half-open interval topology]] with itself. [[Open set]]s 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 <math>\mathbb{S}</math> is the set of points <math>(x,y)</math> such that <math>x+y=0</math>.
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| Consider the [[open covering]] of <math>\mathbb{S}</math> which consists of:
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| # The set of all rectangles <math>(-\infty,x)\times(-\infty,y)</math>, where <math>(x,y)</math> is on the antidiagonal.
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| # The set of all rectangles <math>[x,+\infty)\times[y,+\infty)</math>, where <math>(x,y)</math> is on the antidiagonal.
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| 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.
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| Another way to see that <math>S</math> is not Lindelöf is to note that the antidiagonal defines a closed and [[uncountable]] [[discrete space|discrete]] subspace of <math>S</math>. This subspace is not Lindelöf, and so the whole space cannot be Lindelöf as well (as closed subspaces of Lindelöf spaces are also Lindelöf).
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| The product of a Lindelöf space and a compact space is Lindelöf.
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| == Generalisation ==
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| The following definition generalises the definitions of compact and Lindelöf: a topological space is <math>\kappa</math>''-compact'' (or <math>\kappa</math>''-Lindelöf''), where <math>\kappa</math> is any [[cardinal number|cardinal]], if every open [[cover (topology)|cover]] has a subcover of cardinality ''strictly'' less than <math>\kappa</math>. Compact is then <math>\aleph_0</math>-compact and Lindelöf is then <math>\aleph_1</math>-compact.
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| The ''Lindelöf degree'', or ''Lindelöf number'' <math>l(X)</math>, is the smallest cardinal <math>\kappa</math> such that every open cover of the space <math>X</math> has a subcover of size at most <math>\kappa</math>. In this notation, <math>X</math> is Lindelöf if <math>l(X) = \aleph_0</math>. 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 <math>\kappa</math> such that every open cover of the space <math>X</math> has a subcover of size strictly less than <math>\kappa</math>.<ref>Mary Ellen Rudin, Lectures on set theoretic topology, Conference Board of the Mathematical Sciences, American Mathematical Society, 1975, p. 4, retrievable on Google Books [http://books.google.it/books?id=_LiqC3Y3kmsC&pg=PA4&dq=%22between+compact+and+lindel%C3%B6f%22&hl=it&ei=3SZtTdTGCYu28QP68aGYBQ&sa=X&oi=book_result&ct=book-thumbnail&resnum=1&ved=0CCwQ6wEwAA#v=onepage&q&f=false]</ref> In this latter (and less used sense) the Lindelöf number is the smallest cardinal <math>\kappa</math> such that a topological space <math>X</math> is <math>\kappa</math>-compact. This notion is sometimes also called the ''compactness degree''{{Citation needed|date=February 2011}} of the space <math>X</math>.
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| == See also ==
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| * [[Axioms of countability]]
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| * [[Lindelöf's lemma]]
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| ==Notes==
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| {{reflist}}
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| == References ==
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| {{refbegin}}
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| * Michael Gemignani, ''Elementary Topology'' (ISBN 0-486-66522-4) (see especially section 7.2)
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| * {{Cite book | last1=Steen | first1=Lynn Arthur | author1-link=Lynn Arthur Steen | last2=Seebach | first2=J. Arthur Jr. | author2-link=J. Arthur Seebach, Jr. | title=[[Counterexamples in Topology]] | origyear=1978 | publisher=[[Springer-Verlag]] | location=Berlin, New York | edition=[[Dover Publications|Dover]] reprint of 1978 | isbn=978-0-486-68735-3 | mr=507446 | year=1995 | postscript=<!--None-->}}
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| * {{cite book | author=I. Juhász | title=Cardinal functions in topology - ten years later | publisher=Math. Centre Tracts, Amsterdam | year=1980 | isbn=90-6196-196-3}}
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| * {{cite book | last=Munkres | first=James | author-link=James Munkres | title=Topology, 2nd ed.}}
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| {{refend}}
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| {{DEFAULTSORT:Lindelof space}}
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| [[Category:Compactness (mathematics)]]
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| [[Category:General topology]]
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| [[Category:Properties of topological spaces]]
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| [[Category:Topology]]
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