Fréchet–Urysohn space

{{See also|Sequential space}}

Let $(X,\; \backslash tau)$ be a topological space.

The {{em|sequential closure}} of $S$ in $(X,\; \backslash tau)$ is the set:

$$\backslash begin\{alignat\}\{4\}$$

\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}

where $\backslash operatorname\{scl\}\_X\; S$ or $\backslash operatorname\{scl\}\_\{(X,\; \backslash tau)\}\; S$ may be written if clarity is needed.

A topological space $(X,\; \backslash tau)$ is said to be a {{em|Fréchet–Urysohn space}} if $$\backslash operatorname\{cl\}\_X\; S\; =\; \backslash operatorname\{scl\}\_X\; S$$

for every subset $S\; \backslash subseteq\; X,$ where $\backslash operatorname\{cl\}\_X\; S$ denotes the closure of $S$ in $(X,\; \backslash tau).$

Suppose that $S\; \backslash subseteq\; X$ is any subset of $X.$

A sequence $x\_1,\; x\_2,\; \backslash ldots$ is {{em|eventually in $S$}} if there exists a positive integer $N$ such that $x\_i\; \backslash in\; S$ for all indices $i\; \backslash geq\; N.$

The set $S$ is called {{em|sequentially open}} if every sequence $\backslash left(x\_i\backslash right)\_\{i=1\}^\{\backslash infty\}$ in $X$ that converges to a point of $S$ is eventually in $S$;

Typically, if $X$ is understood then $\backslash operatorname\{scl\}\; S$ is written in place of $\backslash operatorname\{scl\}\_X\; S.$

The set $S$ is called {{em|sequentially closed}} if $S\; =\; \backslash operatorname\{scl\}\_X\; S,$ or equivalently, if whenever $x\_\{\backslash bull\}\; =\; \backslash left(x\_i\backslash right)\_\{i=1\}^\{\backslash infty\}$ is a sequence in $S$ converging to $x,$ then $x$ must also be in $S.$

The complement of a sequentially open set is a sequentially closed set, and vice versa.

Let

$$\backslash begin\{alignat\}\{4\}$$

\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}

denote the set of all sequentially open subsets of $(X,\; \backslash tau),$ where this may be denoted by $\backslash operatorname\{SeqOpen\}\; X$ is the topology $\backslash tau$ is understood.

The set $\backslash operatorname\{SeqOpen\}\; (X,\; \backslash tau)$ is a topology on $X$ that is finer than the original topology $\backslash tau.$

Every open (resp. closed) subset of $X$ is sequentially open (resp. sequentially closed), which implies that $$\backslash tau\; \backslash subseteq\; \backslash operatorname\{SeqOpen\}\; (X,\; \backslash tau).$$

A topological space $X$ is a {{em|strong Fréchet–Urysohn space}} if for every point $x\; \backslash in\; X$ and every sequence $A\_1,\; A\_2,\; \backslash ldots$ of subsets of the space $X$ such that $x\; \backslash in\; \backslash bigcap\_n\; \backslash overline\{A\_n\},$ there exist a sequence $\backslash left(\; a\_i\; \backslash right)\_\{i=1\}^\{\backslash infty\}$ in $X$ such that $a\_i\; \backslash in\; A\_i$ for every $i\; \backslash in\; \backslash mathbb\{N\}$ and $\backslash left(\; a\_i\; \backslash right)\_\{i=1\}^\{\backslash infty\}\; \backslash to\; x$ in $(X,\; \backslash tau).$

The above properties can be expressed as selection principles.

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 {{em|sequential spaces}};

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 $S\; \backslash subseteq\; X,$ knowledge of which sequences in $X$ converge to which point(s) of $X$ (and which do not) is sufficient to {{em|determine whether or not}} $S$ is closed in $X$ (respectively, is sufficient to {{em|determine the closure}} of $S$ in $X$).Of course, if you can determine {{em|all}} 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 {{em|only}} determine whether or not $S$ 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 $S$ 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 {{em|not}} sequential, there exists a subset for which this "test" gives a "false positive."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).

If $(X,\; \backslash tau)$ is a topological space then the following are equivalent:

1. $X$ is a Fréchet–Urysohn space.
2. Definition: $\backslash operatorname\{scl\}\_X\; S\; ~=~\; \backslash operatorname\{cl\}\_X\; S$ for every subset $S\; \backslash subseteq\; X.$
3. $\backslash operatorname\{scl\}\_X\; S\; ~\backslash supseteq~\; \backslash operatorname\{cl\}\_X\; S$ for every subset $S\; \backslash subseteq\; X.$
   • This statement is equivalent to the definition above because $\backslash operatorname\{scl\}\_X\; S\; ~\backslash subseteq~\; \backslash operatorname\{cl\}\_X\; S$ always holds for every $S\; \backslash subseteq\; X.$

4. Every subspace of $X$ is a sequential space.
5. For any subset $S\; \backslash subseteq\; X$ that is {{em|not}} closed in $X$ and {{em|for every}} $x\; \backslash in\; \backslash left(\; \backslash operatorname\{cl\}\_X\; S\; \backslash right)\; \backslash setminus\; S,$ there exists a sequence in $S$ that converges to $x.$
   • Contrast this condition to the following characterization of a sequential space:

   :For any subset $S\; \backslash subseteq\; X$ that is {{em|not}} closed in $X,$ {{em|there exists}} some $x\; \backslash in\; \backslash left(\; \backslash operatorname\{cl\}\_X\; S\; \backslash right)\; \backslash setminus\; S$ for which there exists a sequence in $S$ that converges to $x.$ Arkhangel'skii, A.V. and Pontryagin L.S.,{{pad|1px}} 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 $(X,\; \backslash tau).$

If $(X,\; \backslash tau)$ is a Hausdorff sequential space then $X$ is a Fréchet–Urysohn space if and only if the following condition holds: If $\backslash left(x\_l\backslash right)\_\{l=1\}^\{\backslash infty\}$ is a sequence in $X$ that converge to some $x\; \backslash in\; X$ and if for every $l\; \backslash in\; \backslash N,$ $\backslash left(x\_l^i\backslash right)\_\{i=1\}^\{\backslash infty\}$ is a sequence in $X$ that converges to $x\_l,$ where these hypotheses can be summarized by the following diagram

$$\backslash begin\{alignat\}\{11\}$$

&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}

then there exist strictly increasing maps $\backslash iota,\; \backslash lambda\; :\; \backslash N\; \backslash to\; \backslash N$ such that $\backslash left(x\_\{\backslash lambda(n)\}^\{\backslash iota(n)\}\backslash right)\_\{n=1\}^\{\backslash infty\}\; \backslash to\; x.$

(It suffices to consider only sequences $\backslash left(x\_l\backslash right)\_\{l=1\}^\{\backslash infty\}$ with infinite ranges (i.e. $\backslash left\backslash \{x\_l\; :\; l\; \backslash in\; \backslash N\backslash right\backslash \}$ 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 $\backslash iota,\; \backslash lambda\; :\; \backslash N\; \backslash to\; \backslash N$ with the desired properties is readily verified for this special case (even if $(X,\; \backslash tau)$ is not a Fréchet–Urysohn space).

Every subspace of a Fréchet–Urysohn space is Fréchet–Urysohn.Engelking 1989, Exercise 2.1.H(b)

Every Fréchet–Urysohn space is a sequential space although the opposite implication is not true in general.Engelking 1989, Example 1.6.18{{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}}

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

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 $(X,\; \backslash tau)$ on a finite set $X$ is a Fréchet–Urysohn space.

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

Direct limit of finite-dimensional Euclidean spaces

{{em|The space of finite real sequences}} $\backslash R^\{\backslash infty\}$ is a Hausdorff sequential space that is not Fréchet–Urysohn.

For every integer $n\; \backslash geq\; 1,$ identify $\backslash R^n$ with the set $\backslash R^n\; \backslash times\; \backslash \{\; \backslash left(\; 0,\; 0,\; 0,\; \backslash ldots\; \backslash right)\backslash \}\; =\; \backslash left\backslash \{\; \backslash left(\; x\_1,\; \backslash ldots,\; x\_n,\; 0,\; 0,\; 0,\; \backslash ldots\; \backslash right)\; ~:~x\_1,\; \backslash ldots,\; x\_n\; \backslash in\; \backslash R\; \backslash right\backslash \},$ where the latter is a subset of the space of sequences of real numbers $\backslash R^\{\backslash mathbb\{N\}\};$ explicitly, the elements $\backslash left(\; x\_1,\; \backslash ldots,\; x\_n\; \backslash right)\; \backslash in\; \backslash R^n$ and $\backslash left(\; x\_1,\; \backslash ldots,\; x\_n,\; 0,\; 0,\; 0,\; \backslash ldots\; \backslash right)$ are identified together.

In particular, $\backslash R^n$ can be identified as a subset of $\backslash R^\{n+1\}$ and more generally, as a subset $\backslash R^n\; \backslash subseteq\; \backslash R^\{n+k\}$ for any integer $k\; \backslash geq\; 0.$ Let

$$\backslash begin\{alignat\}\{4\}$$

\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}

Give $\backslash R^\{\backslash infty\}$ its usual topology $\backslash tau,$ in which a subset $S\; \backslash subseteq\; \backslash R^\{\backslash infty\}$ is open (resp. closed) if and only if for every integer $n\; \backslash geq\; 1,$ the set $S\; \backslash cap\; \backslash R^n\; =\; \backslash left\backslash \{\; \backslash left(\; x\_1,\; \backslash ldots,\; x\_n\; \backslash right)\; ~:~\; \backslash left(\; x\_1,\; \backslash ldots,\; x\_n,\; 0,\; 0,\; \backslash ldots\; \backslash right)\; \backslash in\; S\; \backslash right\backslash \}$ is an open (resp. closed) subset of $\backslash R^n$ (with it usual Euclidean topology).

If $v\; \backslash in\; \backslash R^\{\backslash infty\}$ and $v\_\{\backslash bull\}$ is a sequence in $\backslash R^\{\backslash infty\}$ then $v\_\{\backslash bull\}\; \backslash to\; v$ in $\backslash left(\; \backslash R^\{\backslash infty\},\; \backslash tau\; \backslash right)$ if and only if there exists some integer $n\; \backslash geq\; 1$ such that both $v$ and $v\_\{\backslash bull\}$ are contained in $\backslash R^n$ and $v\_\{\backslash bull\}\; \backslash to\; v$ in $\backslash R^n.$

From these facts, it follows that $\backslash left(\backslash R^\{\backslash infty\},\; \backslash tau\backslash right)$ is a sequential space.

For every integer $n\; \backslash geq\; 1,$ let $B\_n$ denote the open ball in $\backslash R^n$ of radius $1/n$ (in the Euclidean norm) centered at the origin.

Let $S\; :=\; \backslash R^\{\backslash infty\}\; \backslash ,\backslash setminus\backslash ,\; \backslash bigcup\_\{n=1\}^\{\backslash infty\}\; B\_n.$

Then the closure of $S$ is $\backslash left(\backslash R^\{\backslash infty\},\; \backslash tau\backslash right)$ is all of $\backslash R^\{\backslash infty\}$ but the origin $(0,\; 0,\; 0,\; \backslash ldots)$ of $\backslash R^\{\backslash infty\}$ does {{em|not}} belong to the sequential closure of $S$ in $\backslash left(\backslash R^\{\backslash infty\},\; \backslash tau\backslash right).$

In fact, it can be shown that

$$\backslash R^\{\backslash infty\}\; =\; \backslash operatorname\{cl\}\_\{\backslash R^\{\backslash infty\}\}\; S\; ~\backslash neq~\; \backslash operatorname\{scl\}\_\{\backslash R^\{\backslash infty\}\}\; S\; =\; \backslash R^\{\backslash infty\}\; \backslash setminus\; \backslash \{(0,\; 0,\; 0,\; \backslash ldots)\backslash \}.$$

This proves that $\backslash left(\backslash R^\{\backslash infty\},\; \backslash tau\backslash right)$ is not a Fréchet–Urysohn space.

Montel DF-spaces

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

The Schwartz space $\backslash mathcal\{S\}\backslash left(\backslash R^n\backslash right)$ and the space of smooth functions $C^\{\backslash infty\}(U)$

The following extensively used spaces are prominent examples of sequential spaces that are not Fréchet–Urysohn spaces.

Let $\backslash mathcal\{S\}\backslash left(\backslash R^n\backslash right)$ denote the Schwartz space and let $C^\{\backslash infty\}(U)$ denote the space of smooth functions on an open subset $U\; \backslash subseteq\; \backslash R^n,$ where both of these spaces have their usual Fréchet space topologies, as defined in the article about distributions.

Both $\backslash mathcal\{S\}\backslash left(\backslash R^n\backslash right)$ and $C^\{\backslash infty\}(U),$ 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{{cite web |title=Topological vector space |author= |date= |website=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."}} normal reflexive barrelled spaces. The strong dual spaces of both $\backslash mathcal\{S\}\backslash left(\backslash R^n\backslash right)$ and $C^\{\backslash infty\}(U)$ are sequential spaces but {{em|neither one}} of these duals is a Fréchet-Urysohn space.Gabriyelyan, Saak [https://arxiv.org/pdf/1702.07867.pdf "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.

• {{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}}

{{reflist|group=note}}

{{reflist}}

• 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