In the field of
topology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
, a Fréchet–Urysohn 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 po ...
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
In topology and related fields of mathematics, a sequential space is a topological space whose topology can be completely characterized by its convergent/divergent sequences. They can be thought of as spaces that satisfy a very weak axiom of counta ...
.
Fréchet–Urysohn spaces are the most general
class of spaces for which
sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...
s suffice to determine all topological properties of subsets of the space.
That is, Fréchet–Urysohn spaces are exactly those spaces for which knowledge of which sequences converge to which limits (and which sequences do not) suffices to completely determine the space's topology.
Every Fréchet–Urysohn space is a sequential space but not conversely.
The space is named after
Maurice Fréchet Maurice may refer to:
People
*Saint Maurice (died 287), Roman legionary and Christian martyr
*Maurice (emperor) or Flavius Mauricius Tiberius Augustus (539–602), Byzantine emperor
* Maurice (bishop of London) (died 1107), Lord Chancellor and L ...
and
Pavel Urysohn
Pavel Samuilovich Urysohn () (February 3, 1898 – August 17, 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 of which ar ...
.
Definitions
Let
be 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 po ...
.
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
A complement is something that completes something else.
Complement may refer specifically to:
The arts
* Complement (music), an interval that, when added to another, spans an octave
** Aggregate complementation, the separation of pitch-clas ...
of a sequentially open set is a sequentially closed set, 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
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
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, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection ...
). 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 resul ...
."
[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
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 resul ...
;" 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
In topology and related fields of mathematics, a sequential space is a topological space whose topology can be completely characterized by its convergent/divergent sequences. They can be thought of as spaces that satisfy a very weak axiom of counta ...
.
- 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
In topology and related fields of mathematics, a sequential space is a topological space whose topology can be completely characterized by its convergent/divergent sequences. They can be thought of as spaces that satisfy a very weak axiom of counta ...
:
: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
In topology and related fields of mathematics, a sequential space is a topological space whose topology can be completely characterized by its convergent/divergent sequences. They can be thought of as spaces that satisfy a very weak axiom of counta ...
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 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
In general topology and related areas of mathematics, the final topology (or coinduced,
strong, colimit, or inductive topology) on a set X, with respect to a family of functions from topological spaces into X, is the finest topology on X that make ...
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 giv ...
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
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 \mat ...
, every
metrizable space
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, \mathcal) is said to be metrizable if there is a metric d : X \times X \to , \infty) s ...
, 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, \mathcal) is said to be metrizable if there is a metric d : X \times X \to , \infty) s ...
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
In functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces.
They are generalizations of Banach spaces ( normed vector spaces that are complete with respect to th ...
) is a
normable space if and only if its
strong dual space
In functional analysis and related areas of mathematics, the strong dual space of a topological vector space (TVS) X is the continuous dual space X^ of X equipped with the strong (dual) topology or the topology of uniform convergence on bounded sub ...
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) 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 Montel is a given name and surname. Notable people with the name include:
Given name
*Montel Vontavious Porter (born 1973), American professional wrestler
*Montel Williams (born 1956), American television personality and television/radio talk show ...
DF-space
In the field of functional analysis, DF-spaces, also written (''DF'')-spaces are locally convex topological vector space having a property that is shared by locally convex metrizable topological vector spaces. They play a considerable part in the ...
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 on ...
and the space of
smooth functions
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 on ...
and let
denote the space of smooth functions on an open subset
where both of these spaces have their usual
Fréchet space
In functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces.
They are generalizations of Banach spaces ( normed vector spaces that are complete with respect to th ...
topologies, as defined in the article about
distributions.
Both
and
as well as the
strong dual space
In functional analysis and related areas of mathematics, the strong dual space of a topological vector space (TVS) X is the continuous dual space X^ of X equipped with the strong (dual) topology or the topology of uniform convergence on bounded sub ...
s of both these of spaces, are
complete nuclear Montel Montel is a given name and surname. Notable people with the name include:
Given name
*Montel Vontavious Porter (born 1973), American professional wrestler
*Montel Williams (born 1956), American television personality and television/radio talk show ...
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 refinement that is locally finite. These spaces were introduced by . Every compact space is paracompact. Every paracompact Hausdorff space is normal ...
normal Normal(s) or The Normal(s) may refer to:
Film and television
* ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson
* ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie
* ''Norma ...
reflexive barrelled space
In functional analysis and related areas of mathematics, a barrelled space (also written barreled space) is a topological vector space (TVS) for which every barrelled set in the space is a neighbourhood for the zero vector.
A barrelled set or a ...
s. 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