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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 ...
X with the property that for every subset S \subseteq X the closure of S in X is identical to the ''sequential'' closure of S in X. 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 (X, \tau) 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 S in (X, \tau) is the set: \begin \operatorname S :&= S := \left\ \end where \operatorname_X S or \operatorname_ S may be written if clarity is needed. A topological space (X, \tau) is said to be a if \operatorname_X S = \operatorname_X S for every subset S \subseteq X, where \operatorname_X S denotes the closure of S in (X, \tau).


Sequentially open/closed sets

Suppose that S \subseteq X is any subset of X. A sequence x_1, x_2, \ldots is if there exists a positive integer N such that x_i \in S for all indices i \geq N. The set S is called if every sequence \left(x_i\right)_^ in X that converges to a point of S is eventually in S; Typically, if X is understood then \operatorname S is written in place of \operatorname_X S. The set S is called if S = \operatorname_X S, or equivalently, if whenever x_ = \left(x_i\right)_^ is a sequence in S converging to x, then x must also be in S. 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 \begin \operatorname (X, \tau) :&= \left\ \\ &= \left\ \\ \end denote the set of all sequentially open subsets of (X, \tau), where this may be denoted by \operatorname X is the topology \tau is understood. The set \operatorname (X, \tau) 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 X that is finer than the original topology \tau. Every open (resp. closed) subset of X is sequentially open (resp. sequentially closed), which implies that \tau \subseteq \operatorname (X, \tau).


Strong Fréchet–Urysohn space

A topological space X is a if for every point x \in X and every sequence A_1, A_2, \ldots of subsets of the space X such that x \in \bigcap_n \overline, there exist a sequence \left( a_i \right)_^ in X such that a_i \in A_i for every i \in \mathbb and \left( a_i \right)_^ \to x in (X, \tau). The above properties can be expressed as selection principles.


Contrast to sequential spaces

Every open subset of X 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 X where for any single given subset S \subseteq X, knowledge of which sequences in X converge to which point(s) of X (and which do not) is sufficient to S is closed in X (respectively, is sufficient to of S in X).Of course, if you can determine of the supersets of S that are closed in X then you can determine the closure of S. So this interpretation assumes that you can determine whether or not S 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 S can be determined without it ever being necessary to consider a subset of X other than S; this is not always possible in non-Fréchet-Urysohn spaces.
Thus sequential spaces are those spaces X for which sequences in X can be used as a "test" to determine whether or not any given subset is open (or equivalently, closed) in X; 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 S is necessarily sequentially open (resp. sequentially closed) so this "test" will never indicate "false" for any set S that really is open (resp. closed).


Characterizations

If (X, \tau) is a topological space then the following are equivalent:
  1. X is a Fréchet–Urysohn space.
  2. Definition: \operatorname_X S ~=~ \operatorname_X S for every subset S \subseteq X.
  3. \operatorname_X S ~\supseteq~ \operatorname_X S for every subset S \subseteq X. * This statement is equivalent to the definition above because \operatorname_X S ~\subseteq~ \operatorname_X S always holds for every S \subseteq X.
  4. Every subspace of X is a sequential space.
  5. For any subset S \subseteq X that is closed in X and x \in \left( \operatorname_X S \right) \setminus S, there exists a sequence in S that converges to x. * Contrast this condition to the following characterization of a sequential space: :For any subset S \subseteq X that is closed in X, some x \in \left( \operatorname_X S \right) \setminus S for which there exists a sequence in S that converges to x. 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 (X, \tau). If (X, \tau) is a Hausdorff sequential space then X is a Fréchet–Urysohn space if and only if the following condition holds: If \left(x_l\right)_^ is a sequence in X that converge to some x \in X and if for every l \in \N, \left(x_l^i\right)_^ is a sequence in X that converges to x_l, where these hypotheses can be summarized by the following diagram

\begin &x_1^1 ~\;~ &x_1^2 ~\;~ &x_1^3 ~\;~ &x_1^4 ~\;~ &x_1^5 ~~ &\ldots ~~ &x_1^i ~~ \ldots ~~ &\to ~~ &x_1 \\ .2ex&x_2^1 ~\;~ &x_2^2 ~\;~ &x_2^3 ~\;~ &x_2^4 ~\;~ &x_2^5 ~~ &\ldots ~~ &x_2^i ~~ \ldots ~~ &\to ~~ &x_2 \\ .2ex&x_3^1 ~\;~ &x_3^2 ~\;~ &x_3^3 ~\;~ &x_3^4 ~\;~ &x_3^5 ~~ &\ldots ~~ &x_3^i ~~ \ldots ~~ &\to ~~ &x_3 \\ .2ex&x_4^1 ~\;~ &x_4^2 ~\;~ &x_4^3 ~\;~ &x_4^4 ~\;~ &x_4^5 ~~ &\ldots ~~ &x_4^i ~~ \ldots ~~ &\to ~~ &x_4 \\ .5ex& & &\;\,\vdots & & & &\;\,\vdots & &\;\,\vdots \\ .5ex&x_l^1 ~\;~ &x_l^2 ~\;~ &x_l^3 ~\;~ &x_l^4 ~\;~ &x_l^5 ~~ &\ldots ~~ &x_l^i ~~ \ldots ~~ &\to ~~ &x_l \\ .5ex& & &\;\,\vdots & & & &\;\,\vdots & &\;\,\vdots \\ & & & & & & & & &\,\downarrow \\ & & & & & & & & ~~ &\;x \\ \end then there exist strictly increasing maps \iota, \lambda : \N \to \N such that \left(x_^\right)_^ \to x. (It suffices to consider only sequences \left(x_l\right)_^ with infinite ranges (i.e. \left\ is infinite) because if it is finite then Hausdorffness implies that it is necessarily eventually constant with value x, in which case the existence of the maps \iota, \lambda : \N \to \N with the desired properties is readily verified for this special case (even if (X, \tau) 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 ...
(X, \tau) is a Fréchet-Urysohn space then \tau is equal to the final topology on X induced by the set \operatorname\left(
, 1 The comma is a punctuation mark that appears in several variants in different languages. Some typefaces render it as a small line, slightly curved or straight, but inclined from the vertical; others give it the appearance of a miniature fille ...
X\right) of all arcs in (X, \tau), which by definition are continuous paths
, 1 The comma is a punctuation mark that appears in several variants in different languages. Some typefaces render it as a small line, slightly curved or straight, but inclined from the vertical; others give it the appearance of a miniature fille ...
\to (X, \tau) 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 (X, \tau) on a finite set X 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) X (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 X^_b 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 X^_b is a normable space.


Sequential spaces that are not Fréchet–Urysohn

Direct limit of finite-dimensional Euclidean spaces \R^ is a Hausdorff sequential space that is not Fréchet–Urysohn. For every integer n \geq 1, identify \R^n with the set \R^n \times \ = \left\, where the latter is a subset of the space of sequences of real numbers \R^; explicitly, the elements \left( x_1, \ldots, x_n \right) \in \R^n and \left( x_1, \ldots, x_n, 0, 0, 0, \ldots \right) are identified together. In particular, \R^n can be identified as a subset of \R^ and more generally, as a subset \R^n \subseteq \R^ for any integer k \geq 0. Let \begin \R^ := \left\ = \bigcup_^ \R^n. \end Give \R^ its usual topology \tau, in which a subset S \subseteq \R^ is open (resp. closed) if and only if for every integer n \geq 1, the set S \cap \R^n = \left\ is an open (resp. closed) subset of \R^n (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 v \in \R^ and v_ is a sequence in \R^ then v_ \to v in \left( \R^, \tau \right) if and only if there exists some integer n \geq 1 such that both v and v_ are contained in \R^n and v_ \to v in \R^n. From these facts, it follows that \left(\R^, \tau\right) is a sequential space. For every integer n \geq 1, let B_n denote the open ball in \R^n of radius 1/n (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 S := \R^ \,\setminus\, \bigcup_^ B_n. Then the closure of S is \left(\R^, \tau\right) is all of \R^ but the origin (0, 0, 0, \ldots) of \R^ does belong to the sequential closure of S in \left(\R^, \tau\right). In fact, it can be shown that \R^ = \operatorname_ S ~\neq~ \operatorname_ S = \R^ \setminus \. This proves that \left(\R^, \tau\right) 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 ...
\mathcal\left(\R^n\right) 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 C^(U) The following extensively used spaces are prominent examples of sequential spaces that are not Fréchet–Urysohn spaces. Let \mathcal\left(\R^n\right) 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 C^(U) denote the space of smooth functions on an open subset U \subseteq \R^n, where both of these spaces have their usual Fréchet space topologies, as defined in the article about distributions. Both \mathcal\left(\R^n\right) and C^(U), 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 \mathcal\left(\R^n\right) and C^(U) 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