View source for Fréchet–Urysohn space

{{Short description|Property of topological space}}

In the field of topology, a '''Fréchet–Urysohn space''' is a [[Topological space|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|sequential space]]. 

The property is named after Maurice Fréchet and [[Biography:Pavel Urysohn|Pavel Urysohn]].

==Definitions==

Let <math>(X, \tau)</math> be a [[Topological space|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 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|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|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|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|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|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 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|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 subspace of a Fréchet–Urysohn space is Fréchet–Urysohn.<ref>Engelking 1989, Exercise 2.1.H(b)</ref>

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

==Examples==

Every [[First-countable space|first-countable space]] is a Fréchet–Urysohn space. Consequently, every [[Second-countable space|second-countable space]], every [[Metrizable space|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|topological vector space]] (TVS) <math>X</math> (for example, a [[Fréchet space]]) is a normable space if and only if its [[Strong dual space|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 functions <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|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=<!--Not stated--> |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|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.&nbsp;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}} <!--{{sfn|Trèves|2006|p=}}-->

{{DEFAULTSORT:Frechet-Urysohn space}}

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