In the field of
topology
Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...
, a Fréchet–Urysohn space is a
topological space
In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
with the property that for every subset
the
closure of
in
is identical to the ''sequential'' closure of
in
Fréchet–Urysohn spaces are a special type of
sequential space.
The property is named after
Maurice Fréchet
Maurice may refer to:
*Maurice (name), a given name and surname, including a list of people with the name
Places
* or Mauritius, an island country in the Indian Ocean
* Maurice, Iowa, a city
* Maurice, Louisiana, a village
* Maurice River, a t ...
and
Pavel Urysohn
Pavel Samuilovich Urysohn (in Russian: ; 3 February, 1898 – 17 August, 1924) was a Soviet mathematician who is best known for his contributions in dimension theory, and for developing Urysohn's metrization theorem and Urysohn's lemma, both ...
.
Definitions
Let
be a
topological space
In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
.
The of
in
is the set:
where
or
may be written if clarity is needed.
A topological space
is said to be a if
for every subset
where
denotes the
closure of
in
Sequentially open/closed sets
Suppose that
is any subset of
A sequence
is if there exists a positive integer
such that
for all indices
The set
is called if every sequence
in
that converges to a point of
is eventually in
;
Typically, if
is understood then
is written in place of
The set
is called if
or equivalently, if whenever
is a sequence in
converging to
then
must also be in
The
complement of a sequentially
open set
In mathematics, an open set is a generalization of an Interval (mathematics)#Definitions_and_terminology, open interval in the real line.
In a metric space (a Set (mathematics), set with a metric (mathematics), distance defined between every two ...
is a sequentially
closed set
In geometry, topology, and related branches of mathematics, a closed set is a Set (mathematics), set whose complement (set theory), complement is an open set. In a topological space, a closed set can be defined as a set which contains all its lim ...
, and vice versa.
Let
denote the set of all sequentially open subsets of
where this may be denoted by
is the topology
is understood.
The set
is a
topology
Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...
on
that is
finer than the original topology
Every open (resp. closed) subset of
is sequentially open (resp. sequentially closed), which implies that
Strong Fréchet–Urysohn space
A topological space
is a if for every point
and every sequence
of subsets of the space
such that
there exist a sequence
in
such that
for every
and
in
The above properties can be expressed as
selection principles.
Contrast to sequential spaces
Every open subset of
is sequentially open and every closed set is sequentially closed.
However, the converses are in general not true.
The spaces for which the converses are true are called ;
that is, a sequential space is a topological space in which every sequentially open subset is necessarily open, or equivalently, it is a space in which every sequentially closed subset is necessarily closed.
Every Fréchet-Urysohn space is a sequential space but there are sequential spaces that are not Fréchet-Urysohn spaces.
Sequential spaces (respectively, Fréchet-Urysohn spaces) can be viewed/interpreted as exactly those spaces
where for any single given subset
knowledge of which sequences in
converge to which point(s) of
(and which do not) is sufficient to
is closed in
(respectively, is sufficient to of
in
).
[Of course, if you can determine of the supersets of that are closed in then you can determine the closure of So this interpretation assumes that you can determine whether or not is closed (and that this is possible with any other subset); said differently, you cannot apply this "test" (of whether a subset is open/closed) to infinitely many subsets simultaneously (e.g. you can not use something akin to the ]axiom of choice
In mathematics, the axiom of choice, abbreviated AC or AoC, is an axiom of set theory. Informally put, the axiom of choice says that given any collection of non-empty sets, it is possible to construct a new set by choosing one element from e ...
). It is in Fréchet-Urysohn spaces that the closure of a set can be determined without it ever being necessary to consider a subset of other than this is not always possible in non-Fréchet-Urysohn spaces.
Thus sequential spaces are those spaces
for which sequences in
can be used as a "test" to determine whether or not any given subset is open (or equivalently, closed) in
; or said differently, sequential spaces are those spaces whose topologies can be completely characterized in terms of sequence convergence.
In any space that is sequential, there exists a subset for which this "test" gives a "
false positive
A false positive is an error in binary classification in which a test result incorrectly indicates the presence of a condition (such as a disease when the disease is not present), while a false negative is the opposite error, where the test resu ...
."
[Although this "test" (which attempts to answer "is this set open (resp. closed)?") could potentially give a "false positive," it can never give a " false negative;" this is because every open (resp. closed) subset is necessarily sequentially open (resp. sequentially closed) so this "test" will never indicate "false" for any set that really is open (resp. closed).]
Characterizations
If
is a topological space then the following are equivalent:
- is a Fréchet–Urysohn space.
- Definition: for every subset
- for every subset
* This statement is equivalent to the definition above because always holds for every
- Every subspace of is a sequential space.
- For any subset that is closed in and there exists a sequence in that converges to
* Contrast this condition to the following characterization of a sequential space:
:For any subset that is closed in some for which there exists a sequence in that converges to
[ Arkhangel'skii, A.V. and Pontryagin L.S., General Topology I, definition 9 p.12]
* This characterization implies that every Fréchet–Urysohn space is a sequential space.
The characterization below shows that from among Hausdorff sequential spaces, Fréchet–Urysohn spaces are exactly those for which a "
cofinal convergent diagonal sequence" can always be found, similar to the
diagonal principal that is used to
characterize topologies in terms of convergent nets. In the following characterization, all convergence is assumed to take place in
If
is a
Hausdorff sequential space then
is a Fréchet–Urysohn space if and only if the following condition holds: If
is a sequence in
that converge to some
and if for every
is a sequence in
that converges to
where these hypotheses can be summarized by the following diagram
then there exist strictly increasing maps
such that
(It suffices to consider only sequences
with infinite ranges (i.e.
is infinite) because if it is finite then Hausdorffness implies that it is necessarily eventually constant with value
in which case the existence of the maps
with the desired properties is readily verified for this special case (even if
is not a Fréchet–Urysohn space).
Properties
Every subspace of a Fréchet–Urysohn space is Fréchet–Urysohn.
Every Fréchet–Urysohn space is a sequential space although the opposite implication is not true in general.
If a
Hausdorff locally convex topological vector space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is als ...
is a Fréchet-Urysohn space then
is equal to the
final topology on
induced by the set
of all
arcs in
which by definition are continuous
paths that are also
topological embedding
In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup.
When some object X is said to be embedded in another object Y, the embedding is g ...
s.
Examples
Every
first-countable space
In topology, a branch of mathematics, a first-countable space is a topological space satisfying the "first axiom of countability". Specifically, a space X is said to be first-countable if each point has a countable neighbourhood basis (local base ...
is a Fréchet–Urysohn space. Consequently, every
second-countable space, every
metrizable space
In topology and related areas of mathematics, a metrizable space is a topological space that is Homeomorphism, homeomorphic to a metric space. That is, a topological space (X, \tau) is said to be metrizable if there is a Metric (mathematics), metr ...
, and every
pseudometrizable space is a Fréchet–Urysohn space. It also follows that every topological space
on a finite set
is a Fréchet–Urysohn space.
Metrizable continuous dual spaces
A
metrizable
In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space. That is, a topological space (X, \tau) is said to be metrizable if there is a metric d : X \times X \to , \infty) suc ...
locally convex topological vector space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is als ...
(TVS)
(for example, a
Fréchet space) is a
normable space
In mathematics, a norm is a function from a real or complex vector space to the non-negative real numbers that behaves in certain ways like the distance from the origin: it commutes with scaling, obeys a form of the triangle inequality, and ze ...
if and only if its
strong dual space is a Fréchet–Urysohn space,
[Gabriyelyan, S.S]
"On topological spaces and topological groups with certain local countable networks
(2014) or equivalently, if and only if
is a normable space.
Sequential spaces that are not Fréchet–Urysohn
Direct limit of finite-dimensional Euclidean spaces
is a Hausdorff sequential space that is not Fréchet–Urysohn.
For every integer
identify
with the set
where the latter is a subset of the
space of sequences of real numbers
explicitly, the elements
and
are identified together.
In particular,
can be identified as a subset of
and more generally, as a subset
for any integer
Let
Give
its usual topology
in which a subset
is open (resp. closed) if and only if for every integer
the set
is an open (resp. closed) subset of
(with it usual
Euclidean topology
In mathematics, and especially general topology, the Euclidean topology is the natural topology induced on n-dimensional Euclidean space \R^n by the Euclidean metric.
Definition
The Euclidean norm on \R^n is the non-negative function \, \cdot ...
).
If
and
is a sequence in
then
in
if and only if there exists some integer
such that both
and
are contained in
and
in
From these facts, it follows that
is a sequential space.
For every integer
let
denote the open ball in
of radius
(in the
Euclidean norm
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces'' ...
) centered at the origin.
Let
Then the closure of
is
is all of
but the origin
of
does belong to the sequential closure of
in
In fact, it can be shown that
This proves that
is not a Fréchet–Urysohn space.
Montel DF-spaces
Every infinite-dimensional
Montel DF-space is a sequential space but a Fréchet–Urysohn space.
The
Schwartz space
In mathematics, Schwartz space \mathcal is the function space of all functions whose derivatives are rapidly decreasing. This space has the important property that the Fourier transform is an automorphism on this space. This property enables o ...
and the space of
smooth function
In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives (''differentiability class)'' it has over its domain.
A function of class C^k is a function of smoothness at least ; t ...
s
The following extensively used spaces are prominent examples of sequential spaces that are not Fréchet–Urysohn spaces.
Let
denote the
Schwartz space
In mathematics, Schwartz space \mathcal is the function space of all functions whose derivatives are rapidly decreasing. This space has the important property that the Fourier transform is an automorphism on this space. This property enables o ...
and let
denote the space of smooth functions on an open subset
where both of these spaces have their usual
Fréchet space topologies, as defined in the article about
distributions.
Both
and
as well as the
strong dual spaces of both these of spaces, are
complete nuclear Montel ultrabornological spaces, which implies that all four of these locally convex spaces are also
paracompact
In mathematics, a paracompact space is a topological space in which every open cover has an open Cover (topology)#Refinement, refinement that is locally finite collection, locally finite. These spaces were introduced by . Every compact space is par ...
normal reflexive barrelled spaces. The strong dual spaces of both
and
are sequential spaces but of these duals is a
Fréchet-Urysohn space.
[Gabriyelyan, Saa]
"Topological properties of Strict LF-spaces and strong duals of Montel Strict LF-spaces"
(2017)[T. Shirai, Sur les Topologies des Espaces de L. Schwartz, Proc. Japan Acad. 35 (1959), 31-36.]
See also
*
*
*
*
*
Notes
Citations
References
* Arkhangel'skii, A.V. and Pontryagin, L.S., ''General Topology I'', Springer-Verlag, New York (1990) .
* Booth, P.I. and Tillotson, A.,
Monoidal closed, cartesian closed and convenient categories of topological spaces' Pacific J. Math., 88 (1980) pp. 35–53.
* Engelking, R., ''General Topology'', Heldermann, Berlin (1989). Revised and completed edition.
* Franklin, S. P.,
Spaces in Which Sequences Suffice, Fund. Math. 57 (1965), 107-115.
* Franklin, S. P.,
Spaces in Which Sequences Suffice II, Fund. Math. 61 (1967), 51-56.
* Goreham, Anthony,
Sequential Convergence in Topological Spaces
* Steenrod, N.E.,
A convenient category of topological spaces', Michigan Math. J., 14 (1967), 133-152.
*
{{DEFAULTSORT:Frechet-Urysohn space
General topology
Properties of topological spaces