In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a Lindelöf space is a
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poin ...
in which every
open cover
In mathematics, and more particularly in set theory, a cover (or covering) of a set X is a collection of subsets of X whose union is all of X. More formally, if C = \lbrace U_\alpha : \alpha \in A \rbrace is an indexed family of subsets U_\alp ...
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 hereditarily Lindelöf space is a topological space such that every subspace of it is Lindelöf. Such a space is sometimes called strongly Lindelöf, but confusingly that terminology is sometimes used with an altogether different meaning.
The term ''hereditarily Lindelöf'' is more common and unambiguous.
Lindelöf spaces are named after the
Finnish mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems.
Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change.
History
On ...
Ernst Leonard Lindelöf.
Properties of Lindelöf spaces
* Every
compact space
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", i ...
, and more generally every
σ-compact space
In mathematics, a topological space is said to be σ-compact if it is the union of countably many compact subspaces.
A space is said to be σ-locally compact if it is both σ-compact and locally compact.
Properties and examples
* Every compact ...
, 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, but not conversely. For example, there are many compact spaces that are not second countable.
* A
metric space
In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general setti ...
is Lindelöf if and only if it is
separable, and if and only if it is
second-countable
In topology, a second-countable space, also called a completely separable space, is a topological space whose topology has a countable base. More explicitly, a topological space T is second-countable if there exists some countable collection \ma ...
.
* Every
regular Lindelöf space is
normal.
* Every
regular Lindelöf space is
paracompact.
* A countable union of Lindelöf subspaces of a topological space is Lindelöf.
* Every closed subspace of a Lindelöf space is Lindelöf. Consequently, every
Fσ set in a Lindelöf space is Lindelöf.
* Arbitrary subspaces of a Lindelöf space need not be Lindelöf.
* The continuous image of a Lindelöf space is Lindelöf.
* The product of a Lindelöf space and a compact space is Lindelöf.
* The product of a Lindelöf space and a
σ-compact space
In mathematics, a topological space is said to be σ-compact if it is the union of countably many compact subspaces.
A space is said to be σ-locally compact if it is both σ-compact and locally compact.
Properties and examples
* Every compact ...
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 is Lindelöf, but the
Sorgenfrey plane is not Lindelöf.
* In a Lindelöf space, every
locally finite family of nonempty subsets is at most countable.
Properties of hereditarily Lindelöf spaces
* A space is hereditarily Lindelöf if and only if every open subspace of it is Lindelöf.
* 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.
* Every
second-countable space is hereditarily Lindelöf.
* Every countable space is hereditarily Lindelöf.
* Every
Suslin space
In the mathematical discipline of general topology, a Polish space is a separable completely metrizable topological space; that is, a space homeomorphic to a complete metric space that has a countable dense subset. Polish spaces are so named beca ...
is hereditarily Lindelöf.
* Every
Radon measure on a hereditarily Lindelöf space is moderated.
Example: the Sorgenfrey plane is not Lindelöf
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 line
In elementary mathematics, a number line is a picture of a graduated straight line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real number to a po ...
under the
half-open interval topology
In mathematics, the lower limit topology or right half-open interval topology is a topology defined on the set \mathbb of real numbers; it is different from the standard topology on \mathbb (generated by the open intervals) and has a number of in ...
with itself.
Open set
In mathematics, open sets are a generalization of open intervals in the real line.
In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are su ...
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
is the set of points
such that
.
Consider the
open covering
In mathematics, and more particularly in set theory, a cover (or covering) of a set X is a collection of subsets of X whose union is all of X. More formally, if C = \lbrace U_\alpha : \alpha \in A \rbrace is an indexed family of subsets U_\alp ...
of
which consists of:
# The set of all rectangles
, where
is on the antidiagonal.
# The set of all rectangles
, where
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
is not Lindelöf is to note that the antidiagonal defines a closed and uncountable discrete space, discrete subspace of
. This subspace is not Lindelöf, and so the whole space cannot be Lindelöf either (as closed subspaces of Lindelöf spaces are also Lindelöf).
Generalisation
The following definition generalises the definitions of compact and Lindelöf: a topological space is
''-compact'' (or
''-Lindelöf''), where
is any
cardinal
Cardinal or The Cardinal may refer to:
Animals
* Cardinal (bird) or Cardinalidae, a family of North and South American birds
**'' Cardinalis'', genus of cardinal in the family Cardinalidae
**'' Cardinalis cardinalis'', or northern cardinal, t ...
, if every open
cover
Cover or covers may refer to:
Packaging
* Another name for a lid
* Cover (philately), generic term for envelope or package
* Album cover, the front of the packaging
* Book cover or magazine cover
** Book design
** Back cover copy, part of copy ...
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 if
. 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
.
[.]
See also
*
Axioms of countability
*
Lindelöf's lemma
Notes
References
* Engelking, Ryszard, ''General Topology'', Heldermann Verlag Berlin, 1989.
*
*
*
* Willard, Stephen. ''General Topology'', Dover Publications (2004)
{{DEFAULTSORT:Lindelof space
Compactness (mathematics)
General topology
Properties of topological spaces