Lindelöf space#Generalisation

• Every compact space, and more generally every σ-compact space, is Lindelöf. In particular, every countable space is Lindelöf.
• A Lindelöf space is compact if and only if it is countably compact.
• Every second-countable space is Lindelöf,Willard, theorem 16.9, p. 111 but not conversely. For example, there are many compact spaces that are not second-countable.
• A metric space is Lindelöf if and only if it is separable, and if and only if it is second-countable.Willard, theorem 16.11, p. 112
• Every regular Lindelöf space is normal.Willard, theorem 16.8, p. 111
• Every regular Lindelöf space is paracompact.{{Cite journal|last=Michael|first=Ernest|date=1953|title=A note on paracompact spaces|journal=Proceedings of the American Mathematical Society|volume=4|issue=5|pages=831–838|doi=10.1090/S0002-9939-1953-0056905-8|doi-access=free|mr=0056905}}
• A countable union of Lindelöf subspaces of a topological space is Lindelöf.
• Every closed subspace of a Lindelöf space is Lindelöf.Willard, theorem 16.6, p. 110 Consequently, every F_σ set in a Lindelöf space is Lindelöf.
• Arbitrary subspaces of a Lindelöf space need not be Lindelöf.{{Cite web|url=https://dantopology.wordpress.com/2012/04/15/examples-of-lindelof-spaces-that-are-not-hereditarily-lindelof/|title=Examples of Lindelof Spaces that are not Hereditarily Lindelof|date=15 April 2012}}
• The continuous image of a Lindelöf space is Lindelöf.Willard, theorem 16.6, p. 110
• The product of a Lindelöf space and a compact space is Lindelöf.{{Cite web|url=https://dantopology.wordpress.com/2011/05/01/the-tube-lemma/|title=The Tube Lemma|date=2 May 2011}}
• The product of a Lindelöf space and a σ-compact space is Lindelöf. This is a corollary to the previous property.
• The product of two Lindelöf spaces need not be Lindelöf. For example, the Sorgenfrey line $S$ is Lindelöf, but the Sorgenfrey plane $S\; \backslash times\; S$ is not Lindelöf.{{Cite web|url=https://dantopology.wordpress.com/2009/09/27/a-note-on-the-sorgenfrey-line|title = A Note on the Sorgenfrey Line|date = 27 September 2009}}
• In a Lindelöf space, every locally finite family of nonempty subsets is at most countable.

• A space is hereditarily Lindelöf if and only if every open subspace of it is Lindelöf.Engelking, 3.8.A(b), p. 194
• Hereditarily Lindelöf spaces are closed under taking countable unions, subspaces, and continuous images.
• A regular Lindelöf space is hereditarily Lindelöf if and only if it is perfectly normal.Engelking, 3.8.A(c), p. 194{{Cite web|url=https://math.stackexchange.com/a/322506/52912|title=General topology - Another question on hereditarily lindelöf space}}
• Every second-countable space is hereditarily Lindelöf.
• Every countable space is hereditarily Lindelöf.
• Every Suslin space is hereditarily Lindelöf.
• Every Radon measure on a hereditarily Lindelöf space is moderated.

The product of Lindelöf spaces is not necessarily Lindelöf. The usual example of this is the Sorgenfrey plane $\backslash mathbb\{S\},$ which is the product of the real line $\backslash Reals$ 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 $\backslash mathbb\{S\}$ is the set of points $(x,\; y)$ such that $x\; +\; y\; =\; 0.$

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

1. The set of all rectangles $(-\backslash infty,\; x)\; \backslash times\; (-\backslash infty,y),$ where $(x,\; y)$ is on the antidiagonal.
2. The set of all rectangles $[x,\; +\backslash infty)\; \backslash times\; [y,+\backslash infty),$ 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 the (uncountably many) sets of item (2) above 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 either (as closed subspaces of Lindelöf spaces are also Lindelöf).

The following definition generalises the definitions of compact and Lindelöf: a topological space is $\backslash kappa$-compact (or $\backslash kappa$-Lindelöf), where $\backslash kappa$ is any cardinal, if every open cover has a subcover of cardinality strictly less than $\backslash kappa$. Compact is then $\backslash aleph\_0$-compact and Lindelöf is then $\backslash aleph\_1$-compact.

The {{visible anchor|Lindelöf degree}}, or Lindelöf number $l(X),$ is the smallest cardinal $\backslash kappa$ such that every open cover of the space $X$ has a subcover of size at most $\backslash kappa.$ In this notation, $X$ is Lindelöf if $l(X)\; =\; \backslash aleph\_0.$ 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 $\backslash kappa$ such that every open cover of the space $X$ has a subcover of size strictly less than $\backslash kappa.$Mary Ellen Rudin, Lectures on set theoretic topology, Conference Board of the Mathematical Sciences, American Mathematical Society, 1975, p. 4, retrievable on Google Books [https://books.google.com/books?id=_LiqC3Y3kmsC&dq=%22between+compact+and+lindel%C3%B6f%22&pg=PA4] In this latter (and less used) sense the Lindelöf number is the smallest cardinal $\backslash kappa$ such that a topological space $X$ is $\backslash kappa$-compact. This notion is sometimes also called the {{visible anchor|compactness degree}} of the space $X.${{cite journal

| last = Hušek | first = Miroslav

| doi = 10.1007/BF01124977 | doi-access = free

| journal = Mathematische Zeitschrift

| mr = 0244947

| pages = 123–126

| title = The class of k-compact spaces is simple

| volume = 110

| year = 1969| issue = 2

| s2cid = 120212653

}}.

• {{annotated link|Axioms of countability}}
• {{annotated link|Lindelöf's lemma}}

{{reflist}}

• Engelking, Ryszard, General Topology, Heldermann Verlag Berlin, 1989. {{ISBN|3-88538-006-4}}
• {{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}}
• {{cite book | last=Munkres | first=James | author-link=James Munkres | title=Topology, 2nd ed.}}
• {{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 | orig-date=1978 | publisher=Springer-Verlag | location=Berlin, New York | edition=Dover reprint of 1978 | isbn=978-0-486-68735-3 | mr=507446 | year=1995}}
• Willard, Stephen. General Topology, Dover Publications (2004) {{ISBN|0-486-43479-6}}

• https://dantopology.wordpress.com/2012/05/03/when-is-a-lindelof-space-normal/

{{DEFAULTSORT:Lindelof space}}

Category:Compactness (mathematics)

Category:General topology

Category:Properties of topological spaces