Lindelöf space
Encyclopedia
In mathematics
, a Lindelöf space is a topological space
in which every open cover has a countable
subcover. The Lindelöf property 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. Such spaces are also known as hereditarily Lindelöf spaces, because all subspaces
of such a space are Lindelöf.
Lindelöf spaces are named for the Finnish
mathematician
Ernst Leonard Lindelöf
.
. 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 space
s. 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 σ-compact space is Lindelöf.
, which is the product of the real line
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.
Consider the open covering of
which consists of:
The thing to notice here is that each rectangle
covers exactly one of the points on the line 
. 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
is not Lindelöf is to note that the line 
defines a closed and uncountable discrete
subspace of
. 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.
-compact (or 
-Lindelöf), where 
is any cardinal
, if every open cover
has a subcover of cardinality strictly less than
. Compact is then 
-compact and Lindelöf is then 
-compact.
The Lindelöf degree, or Lindelöf number
, is the smallest cardinal 
such that every open cover of the space 
has a subcover of size at most 
. In this notation, 
is Lindelöf iff 
. 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 
such that every open cover of the space 
has a subcover of size strictly less than 
. In this latter (and less used sense) the Lindelöf number is the smallest cardinal 
such that a topological space 
is 
-compact. This notion is sometimes also called the compactness degree of the space 
.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a Lindelöf space is a topological space
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
in which every open cover has a countable
Countable set
In mathematics, a countable set is a set with the same cardinality as some subset of the set of natural numbers. A set that is not countable is called uncountable. The term was originated by Georg Cantor...
subcover. The Lindelöf property is a weakening of the more commonly used notion of compactness
Compact space
In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
, which requires the existence of a finite subcover.
A strongly Lindelöf space is a topological space such that every open subset is Lindelöf. Such spaces are also known as hereditarily Lindelöf spaces, because all subspaces
Subspace topology
In topology and related areas of mathematics, a subspace of a topological space X is a subset S of X which is equipped with a natural topology induced from that of X called the subspace topology .- Definition :Given a topological space and a subset S of X, the...
of such a space are Lindelöf.
Lindelöf spaces are named for the Finnish
Finland
Finland , officially the Republic of Finland, is a Nordic country situated in the Fennoscandian region of Northern Europe. It is bordered by Sweden in the west, Norway in the north and Russia in the east, while Estonia lies to its south across the Gulf of Finland.Around 5.4 million people reside...
mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
Ernst Leonard Lindelöf
Ernst Leonard Lindelöf
Ernst Leonard Lindelöf, was a Finnish topologist after whom Lindelöf spaces are named; he was the son of Leonard Lorenz Lindelöf and brother of the philologist Uno Lorenz Lindelöf....
.
Properties of Lindelöf spaces
In general, no implications hold (in either direction) between the Lindelöf property and other compactness properties, such as paracompactnessParacompact 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 ....
. But by the Morita theorem, every regular
Regular space
In topology and related fields of mathematics, a topological space X is called a regular space if every non-empty closed subset C of X and a point p not contained in C admit non-overlapping open neighborhoods. Thus p and C...
Lindelöf space is paracompact. Also, any second-countable space
Second-countable space
In topology, a second-countable space, also called a completely separable space, is a topological space satisfying the second axiom of countability. A space is said to be second-countable if its topology has a countable base...
is a Lindelöf space, but not conversely.
However, the matter is simpler for metric space
Metric space
In mathematics, a metric space is a set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space...
s. A metric space is Lindelöf if and only if it is separable, and if and only if it is second-countable
Second-countable space
In topology, a second-countable space, also called a completely separable space, is a topological space satisfying the second axiom of countability. A space is said to be second-countable if its topology has a countable base...
.
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 σ-compact space is Lindelöf.
Properties of strongly Lindelöf spaces
- Any second-countable spaceSecond-countable spaceIn topology, a second-countable space, also called a completely separable space, is a topological space satisfying the second axiom of countability. A space is said to be second-countable if its topology has a countable base...
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 measureRadon measureIn mathematics , a Radon measure, named after Johann Radon, is a measure on the σ-algebra of Borel sets of a Hausdorff topological space X that is locally finite and inner regular.-Motivation:...
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, which is the product of the real lineReal line
In mathematics, the real line, or real number line is the line whose points are the real numbers. That is, the real line is the set of all real numbers, viewed as a geometric space, namely the Euclidean space of dimension one...
under the half-open interval topology with itself. Open setOpen set
The concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...
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.
Consider the open covering of
which consists of:- The set of all points
, 
with 
- The set of all points
, 
with 
- For each real
, the half-open rectangle 
The thing to notice here is that each rectangle
covers exactly one of the points on the line . 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
is not Lindelöf is to note that the line defines a closed and uncountable discreteDiscrete space
In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points are "isolated" from each other in a certain sense.- Definitions :Given a set X:...
subspace of
. 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-compact (or -Lindelöf), where is any cardinalCardinal number
In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality of sets. The cardinality of a finite set is a natural number – the number of elements in the set. The transfinite cardinal numbers describe the sizes of infinite...
, if every open cover
Cover (topology)
In mathematics, a cover of a set X is a collection of sets whose union contains X as a subset. Formally, ifC = \lbrace U_\alpha: \alpha \in A\rbrace...
has a subcover of cardinality strictly less than
. Compact is then -compact and Lindelöf is then -compact.The Lindelöf degree, or Lindelöf number
, is the smallest cardinal such that every open cover of the space has a subcover of size at most . In this notation, is Lindelöf iff . 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 such that every open cover of the space has a subcover of size strictly less than . In this latter (and less used sense) the Lindelöf number is the smallest cardinal such that a topological space is -compact. This notion is sometimes also called the compactness degree of the space .