Fréchet–Urysohn space

Suppose that ${\displaystyle S\subseteq X}$ is any subset of ${\displaystyle X.}$ A sequence ${\displaystyle x_{1},x_{2},\ldots }$ is eventually in ${\displaystyle S}$ if there exists a positive integer ${\displaystyle N}$ such that ${\displaystyle x_{i}\in S}$ for all indices ${\displaystyle i\geq N.}$

The set ${\displaystyle S}$ is called sequentially open if every sequence ${\displaystyle \left(x_{i}\right)_{i=1}^{\infty }}$ in ${\displaystyle X}$ that converges to a point of ${\displaystyle S}$ is eventually in ${\displaystyle S}$; Typically, if ${\displaystyle X}$ is understood then ${\displaystyle \operatorname {scl} S}$ is written in place of ${\displaystyle \operatorname {scl} _{X}S.}$

The set ${\displaystyle S}$ is called sequentially closed if ${\displaystyle S=\operatorname {scl} _{X}S,}$ or equivalently, if whenever ${\displaystyle x_{\bullet }=\left(x_{i}\right)_{i=1}^{\infty }}$ is a sequence in ${\displaystyle S}$ converging to ${\displaystyle x,}$ then ${\displaystyle x}$ must also be in ${\displaystyle S.}$ The complement of a sequentially open set is a sequentially closed set, and vice versa.

Let

$${\displaystyle {\begin{alignedat}{4}\operatorname {SeqOpen} (X,\tau ):&=\left\{S\subseteq X~:~S{\text{ is sequentially open in }}(X,\tau )\right\}\\&=\left\{S\subseteq X~:~S=\operatorname {SeqInt} _{(X,\tau )}S\right\}\\\end{alignedat}}}$$

denote the set of all sequentially open subsets of ${\displaystyle (X,\tau ),}$ where this may be denoted by ${\displaystyle \operatorname {SeqOpen} X}$ is the topology ${\displaystyle \tau }$ is understood. The set ${\displaystyle \operatorname {SeqOpen} (X,\tau )}$ is a topology on ${\displaystyle X}$ that is finer than the original topology ${\displaystyle \tau .}$

Every open (resp. closed) subset of ${\displaystyle X}$ is sequentially open (resp. sequentially closed), which implies that

$${\displaystyle \tau \subseteq \operatorname {SeqOpen} (X,\tau ).}$$

A topological space ${\displaystyle X}$ is a strong Fréchet–Urysohn space if for every point ${\displaystyle x\in X}$ and every sequence ${\displaystyle A_{1},A_{2},\ldots }$ of subsets of the space ${\displaystyle X}$ such that ${\displaystyle x\in \bigcap _{n}{\overline {A_{n}}},}$ there exist a sequence ${\displaystyle \left(a_{i}\right)_{i=1}^{\infty }}$ in ${\displaystyle X}$ such that ${\displaystyle a_{i}\in A_{i}}$ for every ${\displaystyle i\in \mathbb {N} }$ and ${\displaystyle \left(a_{i}\right)_{i=1}^{\infty }\to x}$ in ${\displaystyle (X,\tau ).}$ The above properties can be expressed as selection principles.

Every open subset of ${\displaystyle X}$ 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 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 ${\displaystyle X}$ where for any single given subset ${\displaystyle S\subseteq X,}$ knowledge of which sequences in ${\displaystyle X}$ converge to which point(s) of ${\displaystyle X}$ (and which do not) is sufficient to determine whether or not ${\displaystyle S}$ is closed in ${\displaystyle X}$ (respectively, is sufficient to determine the closure of ${\displaystyle S}$ in ${\displaystyle X}$).[note 1] Thus sequential spaces are those spaces ${\displaystyle X}$ for which sequences in ${\displaystyle X}$ can be used as a "test" to determine whether or not any given subset is open (or equivalently, closed) in ${\displaystyle X}$; or said differently, sequential spaces are those spaces whose topologies can be completely characterized in terms of sequence convergence. In any space that is not sequential, there exists a subset for which this "test" gives a "false positive."[note 2]

A metrizable locally convex topological vector space (TVS) ${\displaystyle X}$ (for example, a Fréchet space) is a normable space if and only if its strong dual space ${\displaystyle X_{b}^{\prime }}$ is a Fréchet–Urysohn space,[4] or equivalently, if and only if ${\displaystyle X_{b}^{\prime }}$ is a normable space.[5]

Direct limit of finite-dimensional Euclidean spaces

The space of finite real sequences ${\displaystyle \mathbb {R} ^{\infty }}$ is a Hausdorff sequential space that is not Fréchet–Urysohn. For every integer ${\displaystyle n\geq 1,}$ identify ${\displaystyle \mathbb {R} ^{n}}$ with the set ${\displaystyle \mathbb {R} ^{n}\times \{\left(0,0,0,\ldots \right)\}=\left\{\left(x_{1},\ldots ,x_{n},0,0,0,\ldots \right)~:~x_{1},\ldots ,x_{n}\in \mathbb {R} \right\},}$ where the latter is a subset of the space of sequences of real numbers ${\displaystyle \mathbb {R} ^{\mathbb {N} };}$ explicitly, the elements ${\displaystyle \left(x_{1},\ldots ,x_{n}\right)\in \mathbb {R} ^{n}}$ and ${\displaystyle \left(x_{1},\ldots ,x_{n},0,0,0,\ldots \right)}$ are identified together. In particular, ${\displaystyle \mathbb {R} ^{n}}$ can be identified as a subset of ${\displaystyle \mathbb {R} ^{n+1}}$ and more generally, as a subset ${\displaystyle \mathbb {R} ^{n}\subseteq \mathbb {R} ^{n+k}}$ for any integer ${\displaystyle k\geq 0.}$ Let

$${\displaystyle {\begin{alignedat}{4}\mathbb {R} ^{\infty }:=\left\{\left(x_{1},x_{2},\ldots \right)\in \mathbb {R} ^{\mathbb {N} }~:~{\text{ all but finitely many }}x_{i}{\text{ are equal to }}0\right\}=\bigcup _{n=1}^{\infty }\mathbb {R} ^{n}.\end{alignedat}}}$$

Give ${\displaystyle \mathbb {R} ^{\infty }}$ its usual topology ${\displaystyle \tau ,}$ in which a subset ${\displaystyle S\subseteq \mathbb {R} ^{\infty }}$ is open (resp. closed) if and only if for every integer ${\displaystyle n\geq 1,}$ the set ${\displaystyle S\cap \mathbb {R} ^{n}=\left\{\left(x_{1},\ldots ,x_{n}\right)~:~\left(x_{1},\ldots ,x_{n},0,0,\ldots \right)\in S\right\}}$ is an open (resp. closed) subset of ${\displaystyle \mathbb {R} ^{n}}$ (with it usual Euclidean topology). If ${\displaystyle v\in \mathbb {R} ^{\infty }}$ and ${\displaystyle v_{\bullet }}$ is a sequence in ${\displaystyle \mathbb {R} ^{\infty }}$ then ${\displaystyle v_{\bullet }\to v}$ in ${\displaystyle \left(\mathbb {R} ^{\infty },\tau \right)}$ if and only if there exists some integer ${\displaystyle n\geq 1}$ such that both ${\displaystyle v}$ and ${\displaystyle v_{\bullet }}$ are contained in ${\displaystyle \mathbb {R} ^{n}}$ and ${\displaystyle v_{\bullet }\to v}$ in ${\displaystyle \mathbb {R} ^{n}.}$ From these facts, it follows that ${\displaystyle \left(\mathbb {R} ^{\infty },\tau \right)}$ is a sequential space. For every integer ${\displaystyle n\geq 1,}$ let ${\displaystyle B_{n}}$ denote the open ball in ${\displaystyle \mathbb {R} ^{n}}$ of radius ${\displaystyle 1/n}$ (in the Euclidean norm) centered at the origin. Let ${\displaystyle S:=\mathbb {R} ^{\infty }\,\setminus \,\bigcup _{n=1}^{\infty }B_{n}.}$ Then the closure of ${\displaystyle S}$ is ${\displaystyle \left(\mathbb {R} ^{\infty },\tau \right)}$ is all of ${\displaystyle \mathbb {R} ^{\infty }}$ but the origin ${\displaystyle (0,0,0,\ldots )}$ of ${\displaystyle \mathbb {R} ^{\infty }}$ does not belong to the sequential closure of ${\displaystyle S}$ in ${\displaystyle \left(\mathbb {R} ^{\infty },\tau \right).}$ In fact, it can be shown that

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

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

Montel DF-spaces

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

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

The following extensively used spaces are prominent examples of sequential spaces that are not Fréchet–Urysohn spaces. Let ${\displaystyle {\mathcal {S}}\left(\mathbb {R} ^{n}\right)}$ denote the Schwartz space and let ${\displaystyle C^{\infty }(U)}$ denote the space of smooth functions on an open subset ${\displaystyle U\subseteq \mathbb {R} ^{n},}$ where both of these spaces have their usual Fréchet space topologies, as defined in the article about distributions. Both ${\displaystyle {\mathcal {S}}\left(\mathbb {R} ^{n}\right)}$ and ${\displaystyle C^{\infty }(U),}$ 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[6] normal reflexive barrelled spaces. The strong dual spaces of both ${\displaystyle {\mathcal {S}}\left(\mathbb {R} ^{n}\right)}$ and ${\displaystyle C^{\infty }(U)}$ are sequential spaces but neither one of these duals is a Fréchet-Urysohn space.[7][8]