1237: – property of certain mathematical objects (usually in a category) that asserts the existence of a countable set with certain properties. Without such an axiom, such a set might not probably exist.
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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
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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 the (uncountably many) sets of item (2) above are needed.
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1485:
Mary Ellen Rudin, Lectures on set theoretic topology, Conference Board of the
Mathematical Sciences, American Mathematical Society, 1975, p. 4, retrievable on Google Books
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The Lindelöf number as defined above does not distinguish between compact spaces and Lindelöf non-compact spaces. Some authors gave the name
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This subspace is not Lindelöf, and so the whole space cannot be Lindelöf either (as closed subspaces of Lindelöf spaces are also Lindelöf).
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Hereditarily Lindelöf spaces are closed under taking countable unions, subspaces, and continuous images.
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The following definition generalises the definitions of compact and Lindelöf: a topological space is
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is a topological space such that every subspace of it is Lindelöf. Such a space is sometimes called
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A space is hereditarily Lindelöf if and only if every open subspace of it is Lindelöf.
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subcover. The Lindelöf property is a weakening of the more commonly used notion of
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In this latter (and less used) sense the Lindelöf number is the smallest cardinal
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of Lindelöf spaces is not necessarily Lindelöf. The usual example of this is the
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Every closed subspace of a Lindelöf space is Lindelöf. Consequently, every
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The product of two Lindelöf spaces need not be Lindelöf. For example, the
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A countable union of Lindelöf subspaces of a topological space is Lindelöf.
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A regular Lindelöf space is hereditarily Lindelöf if and only if it is
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is not Lindelöf is to note that the antidiagonal defines a closed and
1473:"General topology - Another question on hereditarily lindelöf space"
109:, is Lindelöf. In particular, every countable space is Lindelöf.
1404:"Examples of Lindelof Spaces that are not Hereditarily Lindelof"
183:
The product of a Lindelöf space and a compact space is Lindelöf.
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Arbitrary subspaces of a Lindelöf space need not be Lindelöf.
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is Lindelöf. This is a corollary to the previous property.
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The continuous image of a Lindelöf space is Lindelöf.
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Pages displaying wikidata descriptions as a fallback
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Pages displaying wikidata descriptions as a fallback
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112:A Lindelöf space is compact if and only if it is
1573:Cardinal functions in topology - ten years later
1358:Proceedings of the American Mathematical Society
638:{\displaystyle [x,+\infty )\times [y,+\infty ),}
544:{\displaystyle (-\infty ,x)\times (-\infty ,y),}
252:family of nonempty subsets is at most countable.
282:Every countable space is hereditarily Lindelöf.
296:on a hereditarily Lindelöf space is moderated.
1619:reprint of 1978 ed.). Berlin, New York:
1072:to a different notion: the smallest cardinal
301:Example: the Sorgenfrey plane is not Lindelöf
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1112:has a subcover of size strictly less than
257:Properties of hereditarily Lindelöf spaces
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186:The product of a Lindelöf space and a
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1296:"A note on strongly Lindelöf spaces"
82:Lindelöf spaces are named after the
56:, which requires the existence of a
1575:. Math. Centre Tracts, Amsterdam.
1560:, Heldermann Verlag Berlin, 1989.
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1372:10.1090/S0002-9939-1953-0056905-8
1061:{\displaystyle l(X)=\aleph _{0}.}
130:is Lindelöf if and only if it is
1680:Properties of topological spaces
174:in a Lindelöf space is Lindelöf.
79:is more common and unambiguous.
1441:"A Note on the Sorgenfrey Line"
980:has a subcover of size at most
1353:"A note on paracompact spaces"
1332:Willard, theorem 16.11, p. 112
1155:such that a topological space
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867:-compact and Lindelöf is then
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1341:Willard, theorem 16.8, p. 111
1323:Willard, theorem 16.9, p. 111
330:{\displaystyle \mathbb {S} ,}
96:Properties of Lindelöf spaces
1649:, Dover Publications (2004)
481:{\displaystyle \mathbb {S} }
389:{\displaystyle \mathbb {S} }
355:{\displaystyle \mathbb {R} }
337:which is the product of the
1612:Counterexamples in Topology
1462:Engelking, 3.8.A(c), p. 194
1453:Engelking, 3.8.A(b), p. 194
1303:Technische Universität Graz
887:{\displaystyle \aleph _{1}}
860:{\displaystyle \aleph _{0}}
364:half-open interval topology
248:In a Lindelöf space, every
134:, and if and only if it is
105:, and more generally every
67:hereditarily Lindelöf space
1696:
1503:-compact spaces is simple"
1275:Willard, Def. 16.5, p. 110
1266:Steen & Seebach, p. 19
586:The set of all rectangles
492:The set of all rectangles
1670:Compactness (mathematics)
1508:Mathematische Zeitschrift
940:is the smallest cardinal
289:is hereditarily Lindelöf.
279:is hereditarily Lindelöf.
238:{\displaystyle S\times S}
27:Type of topological space
1497:Hušek, Miroslav (1969).
1351:Michael, Ernest (1953).
1128:{\displaystyle \kappa .}
996:{\displaystyle \kappa .}
684:Another way to see that
1188:{\displaystyle \kappa }
1148:{\displaystyle \kappa }
1085:{\displaystyle \kappa }
953:{\displaystyle \kappa }
833:{\displaystyle \kappa }
801:{\displaystyle \kappa }
778:{\displaystyle \kappa }
755:{\displaystyle \kappa }
677:is on the antidiagonal.
583:is on the antidiagonal.
1607:Seebach, J. Arthur Jr.
1235:Axioms of countability
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1556:Engelking, Ryszard,
1443:. 27 September 2009.
1294:Ganster, M. (1989).
1284:Willard, 16E, p. 114
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1571:I. Juhász (1980).
1522:10.1007/BF01124977
1219:{\displaystyle X.}
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840:. Compact is then
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1168:{\displaystyle X}
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1016:{\displaystyle X}
973:{\displaystyle X}
697:{\displaystyle S}
209:{\displaystyle S}
114:countably compact
73:strongly Lindelöf
40:topological space
16:(Redirected from
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1595:Topology, 2nd ed
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1664:Categories
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