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In the field of [[Topology (mathematics)|topology]], a '''Fréchet–Urysohn space''' is a [[topological space]] <math>X</math> with the property that for every subset <math>S \subseteq X</math> the [[Closure (topology)|closure]] of <math>S</math> in <math>X</math> is identical to the ''sequential'' closure of <math>S</math> in <math>X.</math>
Fréchet–Urysohn spaces are a special type of [[sequential space]].

Fréchet–Urysohn spaces are the most general [[Class (set theory)|class]] of spaces for which [[sequence]]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]] and [[Pavel Urysohn]].

==Definitions==
{{See also|Sequential space}}

Let <math>(X, \tau)</math> be a [[topological space]].
The '''{{em|sequential closure}}''' of <math>S</math> in <math>(X, \tau)</math> is the set:
<math display=block>\begin{alignat}{4}
\operatorname{scl} S
:&= [ S]_{\operatorname{seq}}
:= \left\{ x \in X ~:~ \text{ there exists a sequence } s_{\bull} = \left(s_i\right)_{i=1}^{\infty} \subseteq S \text{ in } S \text{ such that } s_{\bull} \to x \text{ in } (X, \tau) \right\}
\end{alignat}</math>

where <math>\operatorname{scl}_X S</math> or <math>\operatorname{scl}_{(X, \tau)} S</math> may be written if clarity is needed.

A topological space <math>(X, \tau)</math> is said to be a '''{{em|Fréchet–Urysohn space}}''' if <math display=block>\operatorname{cl}_X S = \operatorname{scl}_X S</math>

for every subset <math>S \subseteq X,</math> where <math>\operatorname{cl}_X S</math> denotes the [[Closure (topology)|closure]] of <math>S</math> in <math>(X, \tau).</math>

===Sequentially open/closed sets===

Suppose that <math>S \subseteq X</math> is any subset of <math>X.</math>
A sequence <math>x_1, x_2, \ldots</math> is '''{{em|eventually in <math>S</math>}}''' if there exists a positive integer <math>N</math> such that <math>x_i \in S</math> for all indices <math>i \geq N.</math>

The set <math>S</math> is called '''{{em|sequentially open}}''' if every sequence <math>\left(x_i\right)_{i=1}^{\infty}</math> in <math>X</math> that converges to a point of <math>S</math> is eventually in <math>S</math>;
Typically, if <math>X</math> is understood then <math>\operatorname{scl} S</math> is written in place of <math>\operatorname{scl}_X S.</math>

The set <math>S</math> is called '''{{em|sequentially closed}}''' if <math>S = \operatorname{scl}_X S,</math> or equivalently, if whenever <math>x_{\bull} = \left(x_i\right)_{i=1}^{\infty}</math> is a sequence in <math>S</math> converging to <math>x,</math> then <math>x</math> must also be in <math>S.</math>
The [[Complement (set theory)|complement]] of a sequentially open set is a sequentially closed set, and vice versa.

Let
<math display=block>\begin{alignat}{4}
\operatorname{SeqOpen} (X, \tau)
:&= \left\{ S \subseteq X ~:~ S \text{ is sequentially open in } (X, \tau) \right\} \\
&= \left\{ S \subseteq X ~:~ S = \operatorname{SeqInt}_{(X, \tau)} S \right\} \\
\end{alignat}</math>

denote the set of all sequentially open subsets of <math>(X, \tau),</math> where this may be denoted by <math>\operatorname{SeqOpen} X</math> is the topology <math>\tau</math> is understood.
The set <math>\operatorname{SeqOpen} (X, \tau)</math> is a [[Topology (structure)|topology]] on <math>X</math> that is [[Comparison of topologies|finer]] than the original topology <math>\tau.</math>
Every open (resp. closed) subset of <math>X</math> is sequentially open (resp. sequentially closed), which implies that <math display=block>\tau \subseteq \operatorname{SeqOpen} (X, \tau).</math>

===Strong Fréchet–Urysohn space===

A topological space <math>X</math> is a '''{{em|strong Fréchet–Urysohn space}}''' if for every point <math>x \in X</math> and every sequence <math>A_1, A_2, \ldots</math> of subsets of the space <math>X</math> such that <math>x \in \bigcap_n \overline{A_n},</math> there exist a sequence <math>\left( a_i \right)_{i=1}^{\infty}</math> in <math>X</math> such that <math>a_i \in A_i</math> for every <math>i \in \mathbb{N}</math> and <math>\left( a_i \right)_{i=1}^{\infty} \to x</math> in <math>(X, \tau).</math>
The above properties can be expressed as [[selection principle]]s.

===Contrast to sequential spaces===

Every open subset of <math>X</math> 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 '''{{em|[[sequential space]]s}}''';
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 <math>X</math> where for any single given subset <math>S \subseteq X,</math> knowledge of which sequences in <math>X</math> converge to which point(s) of <math>X</math> (and which do not) is sufficient to '''{{em|determine whether or not}}''' <math>S</math> is closed in <math>X</math> (respectively, is sufficient to '''{{em|determine the closure}}''' of <math>S</math> in <math>X</math>).<ref group=note>Of course, if you can determine {{em|all}} of the supersets of <math>S</math> that are closed in <math>X</math> then you can determine the closure of <math>S.</math> So this interpretation assumes that you can {{em|only}} determine whether or not <math>S</math> is closed (and that this is {{em|not}} 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]]). It is in Fréchet-Urysohn spaces that the closure of a set <math>S</math> can be determined without it ever being necessary to consider a subset of <math>X</math> other than <math>S;</math> this is not always possible in non-Fréchet-Urysohn spaces.</ref>
Thus sequential spaces are those spaces <math>X</math> for which sequences in <math>X</math> can be used as a "test" to determine whether or not any given subset is open (or equivalently, closed) in <math>X</math>; or said differently, sequential spaces are those spaces whose topologies can be completely characterized in terms of sequence convergence.
In any space that is {{em|not}} sequential, there exists a subset for which this "test" gives a "[[false positive]]."<ref group=note>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 <math>S</math> is necessarily sequentially open (resp. sequentially closed) so this "test" will never indicate "false" for any set <math>S</math> that really is open (resp. closed).</ref>

==Characterizations==

If <math>(X, \tau)</math> is a topological space then the following are equivalent:

<ol>
<li><math>X</math> is a Fréchet–Urysohn space.</li>
<li>Definition: <math>\operatorname{scl}_X S ~=~ \operatorname{cl}_X S</math> for every subset <math>S \subseteq X.</math></li>
<li><math>\operatorname{scl}_X S ~\supseteq~ \operatorname{cl}_X S</math> for every subset <math>S \subseteq X.</math>
* This statement is equivalent to the definition above because <math>\operatorname{scl}_X S ~\subseteq~ \operatorname{cl}_X S</math> always holds for every <math>S \subseteq X.</math></li>
<li>Every subspace of <math>X</math> is a [[sequential space]].</li>
<li>For any subset <math>S \subseteq X</math> that is {{em|not}} closed in <math>X</math> and '''{{em|for every}}''' <math>x \in \left( \operatorname{cl}_X S \right) \setminus S,</math> there exists a sequence in <math>S</math> that converges to <math>x.</math>
* Contrast this condition to the following characterization of a [[sequential space]]:
:For any subset <math>S \subseteq X</math> that is {{em|not}} closed in <math>X,</math> '''{{em|there exists}}''' some <math>x \in \left( \operatorname{cl}_X S \right) \setminus S</math> for which there exists a sequence in <math>S</math> that converges to <math>x.</math><ref name="Arkhangel'skii, A.V. and Pontryagin L.S."> Arkhangel'skii, A.V. and Pontryagin L.S.,{{pad|1px}} General Topology I, definition 9 p.12</ref>
* This characterization implies that every Fréchet–Urysohn space is a sequential space.
</li>
</ol>

The characterization below shows that from among Hausdorff sequential spaces, Fréchet–Urysohn spaces are exactly those for which a "[[Cofinal set|cofinal]] convergent diagonal sequence" can always be found, similar to the [[Axiomatic foundations of topological spaces#Diagonal principal|diagonal principal]] that is used to [[Axiomatic foundations of topological spaces#Convergent net characterization|characterize topologies in terms of convergent nets]]. In the following characterization, all convergence is assumed to take place in <math>(X, \tau).</math>

If <math>(X, \tau)</math> is a [[Hausdorff space|Hausdorff]] [[sequential space]] then <math>X</math> is a Fréchet–Urysohn space if and only if the following condition holds: If <math>\left(x_l\right)_{l=1}^{\infty}</math> is a sequence in <math>X</math> that converge to some <math>x \in X</math> and if for every <math>l \in \N,</math> <math>\left(x_l^i\right)_{i=1}^{\infty}</math> is a sequence in <math>X</math> that converges to <math>x_l,</math> where these hypotheses can be summarized by the following diagram<br /><br />
<math display=block>\begin{alignat}{11}
&x_1^1 ~\;~ &x_1^2 ~\;~ &x_1^3 ~\;~ &x_1^4 ~\;~ &x_1^5 ~~ &\ldots ~~ &x_1^i ~~ \ldots ~~ &\to ~~ &x_1 \\[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 \\[1.2ex]
&x_3^1 ~\;~ &x_3^2 ~\;~ &x_3^3 ~\;~ &x_3^4 ~\;~ &x_3^5 ~~ &\ldots ~~ &x_3^i ~~ \ldots ~~ &\to ~~ &x_3 \\[1.2ex]
&x_4^1 ~\;~ &x_4^2 ~\;~ &x_4^3 ~\;~ &x_4^4 ~\;~ &x_4^5 ~~ &\ldots ~~ &x_4^i ~~ \ldots ~~ &\to ~~ &x_4 \\[0.5ex]
& & &\;\,\vdots & & & &\;\,\vdots & &\;\,\vdots \\[0.5ex]
&x_l^1 ~\;~ &x_l^2 ~\;~ &x_l^3 ~\;~ &x_l^4 ~\;~ &x_l^5 ~~ &\ldots ~~ &x_l^i ~~ \ldots ~~ &\to ~~ &x_l \\[0.5ex]
& & &\;\,\vdots & & & &\;\,\vdots & &\;\,\vdots \\
& & & & & & & & &\,\downarrow \\
& & & & & & & & ~~ &\;x \\
\end{alignat}</math>
then there exist strictly increasing maps <math>\iota, \lambda : \N \to \N</math> such that <math>\left(x_{\lambda(n)}^{\iota(n)}\right)_{n=1}^{\infty} \to x.</math>

(It suffices to consider only sequences <math>\left(x_l\right)_{l=1}^{\infty}</math> with infinite ranges (i.e. <math>\left\{x_l : l \in \N\right\}</math> is infinite) because if it is finite then Hausdorffness implies that it is necessarily eventually constant with value <math>x,</math> in which case the existence of the maps <math>\iota, \lambda : \N \to \N</math> with the desired properties is readily verified for this special case (even if <math>(X, \tau)</math> is not a Fréchet–Urysohn space).

==Properties==

Every Fréchet–Urysohn space is a sequential space although the opposite implication is not true in general.<ref>Engelking 1989, Example 1.6.18</ref><ref>{{cite web|last=Ma|first=Dan|title=A note about the Arens' space|date=19 August 2010|url=http://dantopology.wordpress.com/2010/08/18/a-note-about-the-arens-space/|accessdate=1 August 2013}}</ref>

If a [[Hausdorff space|Hausdorff]] [[Locally convex topological vector space|locally convex]] [[topological vector space]] <math>(X, \tau)</math> is a Fréchet-Urysohn space then <math>\tau</math> is equal to the [[final topology]] on <math>X</math> induced by the set <math>\operatorname{Arc}\left([0, 1]; X\right)</math> of all [[Arc (topology)|arcs]] in <math>(X, \tau),</math> which by definition are continuous [[Path (mathematics)|paths]] <math>[0, 1] \to (X, \tau)</math> that are also [[topological embedding]]s.

==Examples==

Every [[first-countable space]] is a Fréchet–Urysohn space. Consequently, every [[second-countable space]], every [[metrizable space]], and every [[pseudometrizable space]] is a Fréchet–Urysohn space. It also follows that every topological space <math>(X, \tau)</math> on a finite set <math>X</math> is a Fréchet–Urysohn space.

===Metrizable continuous dual spaces===

A [[Metrizable topological vector space|metrizable]] [[Locally convex topological vector space|locally convex]] [[topological vector space]] (TVS) <math>X</math> (for example, a [[Fréchet space]]) is a [[normable space]] if and only if its [[strong dual space]] <math>X^{\prime}_b</math> is a Fréchet–Urysohn space,<ref name="Gabriyelyan 2014">Gabriyelyan, S.S. [https://arxiv.org/pdf/1412.1497.pdf "On topological spaces and topological groups with certain local countable networks] (2014)</ref> or equivalently, if and only if <math>X^{\prime}_b</math> is a normable space.{{sfn|Trèves|2006|p=201}}

===Sequential spaces that are not Fréchet–Urysohn===

'''Direct limit of finite-dimensional Euclidean spaces'''

'''{{em|The [[space of finite real sequences]]}}''' <math>\R^{\infty}</math> is a Hausdorff sequential space that is not Fréchet–Urysohn.
For every integer <math>n \geq 1,</math> identify <math>\R^n</math> with the set <math>\R^n \times \{ \left( 0, 0, 0, \ldots \right)\} = \left\{ \left( x_1, \ldots, x_n, 0, 0, 0, \ldots \right) ~:~x_1, \ldots, x_n \in \R \right\},</math> where the latter is a subset of the [[space of sequences]] of real numbers <math>\R^{\mathbb{N}};</math> explicitly, the elements <math>\left( x_1, \ldots, x_n \right) \in \R^n</math> and <math>\left( x_1, \ldots, x_n, 0, 0, 0, \ldots \right)</math> are identified together.
In particular, <math>\R^n</math> can be identified as a subset of <math>\R^{n+1}</math> and more generally, as a subset <math>\R^n \subseteq \R^{n+k}</math> for any integer <math>k \geq 0.</math> Let
<math display=block>\begin{alignat}{4}
\R^{\infty}
:= \left\{ \left( x_1, x_2, \ldots \right) \in \R^{\mathbb{N}} ~:~ \text{ all but finitely many } x_i \text{ are equal to } 0 \right\}
= \bigcup_{n=1}^{\infty} \R^n.
\end{alignat}</math>
Give <math>\R^{\infty}</math> its usual topology <math>\tau,</math> in which a subset <math>S \subseteq \R^{\infty}</math> is open (resp. closed) if and only if for every integer <math>n \geq 1,</math> the set <math>S \cap \R^n = \left\{ \left( x_1, \ldots, x_n \right) ~:~ \left( x_1, \ldots, x_n, 0, 0, \ldots \right) \in S \right\}</math> is an open (resp. closed) subset of <math>\R^n</math> (with it usual [[Euclidean topology]]).
If <math>v \in \R^{\infty}</math> and <math>v_{\bull}</math> is a sequence in <math>\R^{\infty}</math> then <math>v_{\bull} \to v</math> in <math>\left( \R^{\infty}, \tau \right)</math> if and only if there exists some integer <math>n \geq 1</math> such that both <math>v</math> and <math>v_{\bull}</math> are contained in <math>\R^n</math> and <math>v_{\bull} \to v</math> in <math>\R^n.</math>
From these facts, it follows that <math>\left(\R^{\infty}, \tau\right)</math> is a sequential space.
For every integer <math>n \geq 1,</math> let <math>B_n</math> denote the open ball in <math>\R^n</math> of radius <math>1/n</math> (in the [[Euclidean norm]]) centered at the origin.
Let <math>S := \R^{\infty} \,\setminus\, \bigcup_{n=1}^{\infty} B_n.</math>
Then the closure of <math>S</math> is <math>\left(\R^{\infty}, \tau\right)</math> is all of <math>\R^{\infty}</math> but the origin <math>(0, 0, 0, \ldots)</math> of <math>\R^{\infty}</math> does {{em|not}} belong to the sequential closure of <math>S</math> in <math>\left(\R^{\infty}, \tau\right).</math>
In fact, it can be shown that
<math display=block>\R^{\infty} = \operatorname{cl}_{\R^{\infty}} S ~\neq~ \operatorname{scl}_{\R^{\infty}} S = \R^{\infty} \setminus \{(0, 0, 0, \ldots)\}.</math>
This proves that <math>\left(\R^{\infty}, \tau\right)</math> is not a Fréchet–Urysohn space.

'''Montel DF-spaces'''

Every infinite-dimensional [[Montel space|Montel]] [[DF-space]] is a sequential space but {{em|not}} a Fréchet–Urysohn space.

'''The [[Schwartz space]] <math>\mathcal{S}\left(\R^n\right)</math> and the space of [[smooth function]]s <math>C^{\infty}(U)</math>'''

The following extensively used spaces are prominent examples of sequential spaces that are not Fréchet–Urysohn spaces.
Let <math>\mathcal{S}\left(\R^n\right)</math> denote the [[Schwartz space]] and let <math>C^{\infty}(U)</math> denote the space of smooth functions on an open subset <math>U \subseteq \R^n,</math> where both of these spaces have their usual [[Fréchet space]] topologies, as defined in the article about [[Distribution (mathematics)|distribution]]s.
Both <math>\mathcal{S}\left(\R^n\right)</math> and <math>C^{\infty}(U),</math> as well as the [[strong dual space]]s of both these of spaces, are [[Complete topological vector space|complete]] [[Nuclear space|nuclear]] [[Montel space|Montel]] [[Ultrabornological space|ultrabornological]] spaces, which implies that all four of these locally convex spaces are also [[Paracompact space|paracompact]]<ref name="Encyc. Math TVS">{{cite web |title=Topological vector space |author= |date= |website=Encyclopedia of Mathematics |publisher=Encyclopedia of Mathematics |access-date=September 6, 2020 |url=https://encyclopediaofmath.org/wiki/Topological_vector_space |quote="It is a Montel space, hence paracompact, and so normal."}}</ref> [[Normal space|normal]] [[Reflexive space|reflexive]] [[barrelled space]]s. The strong dual spaces of both <math>\mathcal{S}\left(\R^n\right)</math> and <math>C^{\infty}(U)</math> are sequential spaces but {{em|neither one}} of these duals is a [[Fréchet-Urysohn space]].<ref name="Gabriyelyan 2017">Gabriyelyan, Saak [https://arxiv.org/pdf/1702.07867.pdf "Topological properties of Strict LF-spaces and strong duals of Montel Strict LF-spaces"] (2017)</ref><ref name="Shirai 1959">T. Shirai, Sur les Topologies des Espaces de L. Schwartz, Proc. Japan Acad. 35 (1959), 31-36.</ref>

==See also==

* {{annotated link|Axiom of countability}}
* {{annotated link|First-countable space}}
* {{annotated link|Limit of a sequence}}
* {{annotated link|Sequence covering map}}
* {{annotated link|Sequential space}}

==Notes==

{{reflist|group=note}}

==Citations==

{{reflist}}

==References==

* Arkhangel'skii, A.V. and Pontryagin, L.S., ''General Topology I'', Springer-Verlag, New York (1990) {{isbn|3-540-18178-4}}.
* Booth, P.I. and Tillotson, A., ''[http://projecteuclid.org/euclid.pjm/1102779712 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., "[http://matwbn.icm.edu.pl/ksiazki/fm/fm57/fm5717.pdf Spaces in Which Sequences Suffice]", Fund. Math. '''57''' (1965), 107-115.
* Franklin, S. P., "[http://matwbn.icm.edu.pl/ksiazki/fm/fm61/fm6115.pdf Spaces in Which Sequences Suffice II]", Fund. Math. '''61''' (1967), 51-56.
* Goreham, Anthony, "[https://arxiv.org/abs/math/0412558 Sequential Convergence in Topological Spaces]"
* Steenrod, N.E., ''[http://projecteuclid.org/euclid.mmj/1028999711 A convenient category of topological spaces]'', Michigan Math. J., 14 (1967), 133-152.
* {{Trèves François Topological vector spaces, distributions and kernels}} 

{{DEFAULTSORT:Frechet-Urysohn space}}

[[Category:General topology]]
[[Category:Properties of topological spaces]]

Fréchet–Urysohn space - Revision history

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wikipedia>Mgkrupa: Reworded and changed to a cleaner notation