Fréchet–Urysohn space

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In the field of topology, a Fréchet–Urysohn space is a topological space ${\displaystyle X}$ with the property that for every subset ${\displaystyle S\subseteq X}$ the closure of ${\displaystyle S}$ in ${\displaystyle X}$ is identical to the sequential closure of ${\displaystyle S}$ in ${\displaystyle X.}$ Fréchet–Urysohn spaces are a special type of sequential space.

Fréchet–Urysohn spaces are the most general class of spaces for which sequences 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[]

Let ${\displaystyle (X,\tau )}$ be a topological space. The sequential closure of ${\displaystyle S}$ in ${\displaystyle (X,\tau )}$ is the set:

where ${\displaystyle \operatorname {SeqCl} _{X}S}$ or ${\displaystyle \operatorname {SeqCl} _{(X,\tau )}S}$ may be written if clarity is needed.

A topological space ${\displaystyle (X,\tau )}$ is said to be a Fréchet–Urysohn space if

for every subset ${\displaystyle S\subseteq X,}$ where ${\displaystyle \operatorname {Cl} _{X}S}$ denotes the closure of ${\displaystyle S}$ in ${\displaystyle (X,\tau ).}$

Sequentially open/closed sets[]

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 ${\displaystyle \operatorname {SeqCl} S}$ is written in place of ${\displaystyle \operatorname {SeqCl} _{X}S.}$

The set ${\displaystyle S}$ is called sequentially closed if ${\displaystyle S=\operatorname {SeqCl} _{X}S,}$ 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

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

Strong Fréchet–Urysohn space[]

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.

Contrast to sequential spaces[]

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 (resp. 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 ${\displaystyle X}$ (resp. suffices 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]}

Characterizations[]

If ${\displaystyle (X,\tau )}$ is a topological space then the following are equivalent:

1. ${\displaystyle X}$ is a Fréchet–Urysohn space.
2. Definition: ${\displaystyle \operatorname {SeqCl} _{X}S~=~\operatorname {Cl} _{X}S}$ for every subset ${\displaystyle S\subseteq X.}$
3. ${\displaystyle \operatorname {SeqCl} _{X}S~\supseteq ~\operatorname {Cl} _{X}S}$ for every subset ${\displaystyle S\subseteq X.}$
   • This statement is equivalent to the definition above because ${\displaystyle \operatorname {SeqCl} _{X}S~\subseteq ~\operatorname {Cl} _{X}S}$ always holds for every ${\displaystyle S\subseteq X.}$
4. Every subspace of ${\displaystyle X}$ is a sequential space.
5. For any subset ${\displaystyle S\subseteq X}$ that is not closed in ${\displaystyle X}$ and for every ${\displaystyle x\in \left(\operatorname {Cl} _{X}S\right)\setminus S,}$ there exists a sequence in ${\displaystyle S}$ that converges to ${\displaystyle x.}$
   • Contrast this condition to the following characterization of a sequential space:

   For any subset ${\displaystyle S\subseteq X}$ that is not closed in ${\displaystyle X,}$ there exists some ${\displaystyle x\in \left(\operatorname {Cl} _{X}S\right)\setminus S}$ for which there exists a sequence in ${\displaystyle S}$ that converges to ${\displaystyle x.}$^[1]

   • 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. This "diagonal sequence" can also be seen as a topological analog that is similar in form to the diagonal sequence that appears in diagonalization arguments, except that some sequences ${\displaystyle x_{l}^{\bullet }}$ might be "skipped" by the diagonal sequence. In the following characterization, all convergence is assumed to take place in ${\displaystyle (X,\tau ).}$

If ${\displaystyle (X,\tau )}$ is a Hausdorff sequential space then the following are equivalent:

1. ${\displaystyle X}$ is a Fréchet–Urysohn space.
2. If ${\displaystyle x_{\bullet }=\left(x_{l}\right)_{l=1}^{\infty }\subseteq X}$ is a sequence with an infinite image (or "range") that converge to some ${\displaystyle x\in X}$ and if for every ${\displaystyle l\in \mathbb {N} ,}$ ${\displaystyle x_{l}^{\bullet }=\left(x_{l}^{i}\right)_{i=1}^{\infty }\subseteq X}$ is a sequence that converges to ${\displaystyle x_{l},}$ where these hypotheses can be summarized by the following diagram
   
   
   $${\displaystyle {\begin{alignedat}{11}&x_{1}^{1}~\;~&x_{1}^{2}~\;~&x_{1}^{3}~\;~&x_{1}^{4}~\;~&x_{1}^{5}~~&\ldots ~~&x_{1}^{i}~~\ldots ~~&\to ~~&x_{1}\\[1.2ex]&x_{2}^{1}~\;~&x_{2}^{2}~\;~&x_{2}^{3}~\;~&x_{2}^{4}~\;~&x_{2}^{5}~~&\ldots ~~&x_{2}^{i}~~\ldots ~~&\to ~~&x_{2}\\[1.2ex]&x_{3}^{1}~\;~&x_{3}^{2}~\;~&x_{3}^{3}~\;~&x_{3}^{4}~\;~&x_{3}^{5}~~&\ldots ~~&x_{3}^{i}~~\ldots ~~&\to ~~&x_{3}\\[1.2ex]&x_{4}^{1}~\;~&x_{4}^{2}~\;~&x_{4}^{3}~\;~&x_{4}^{4}~\;~&x_{4}^{5}~~&\ldots ~~&x_{4}^{i}~~\ldots ~~&\to ~~&x_{4}\\[0.5ex]&&&\;\,\vdots &&&&\;\,\vdots &&\;\,\vdots \\[0.5ex]&x_{l}^{1}~\;~&x_{l}^{2}~\;~&x_{l}^{3}~\;~&x_{l}^{4}~\;~&x_{l}^{5}~~&\ldots ~~&x_{l}^{i}~~\ldots ~~&\to ~~&x_{l}\\[0.5ex]&&&\;\,\vdots &&&&\;\,\vdots &&\;\,\vdots \\&&&&&&&&&\,\downarrow \\&&&&&&&&~~&\;x\\\end{alignedat}}}$$

   
   then there exist maps ${\displaystyle \iota ,\lambda :\mathbb {N} \to \mathbb {N} }$ with ${\displaystyle \lambda }$ strictly increasing such that ${\displaystyle \left(x_{\lambda (n)}^{\iota (n)}\right)_{n=1}^{\infty }\to x.}$
   • Because a sequence in ${\displaystyle X}$ is by definition just a map of the form ${\displaystyle \mathbb {N} \to X,}$ the sequence ${\displaystyle x_{\bullet }=\left(x_{l}\right)_{l=1}^{\infty }}$ "having an infinite image" means precisely that the set ${\displaystyle \operatorname {Im} x_{\bullet }:=\left\{x_{l}:l\in \mathbb {N} \right\}}$ is infinite. If ${\displaystyle x_{\bullet }}$ does not have this property then it is necessarily eventually constant with value ${\displaystyle x,}$ in which case the existence of the maps ${\displaystyle \iota ,\lambda :\mathbb {N} \to \mathbb {N} }$ with the desired properties is readily verified for this special case (even if ${\displaystyle (X,\tau )}$ is not a Fréchet–Urysohn space).
3. Statement (2) but with the additional requirement that ${\displaystyle \iota :\mathbb {N} \to \mathbb {N} }$ also be strictly increasing.
   • In short, this condition guarantees that if ${\displaystyle x_{\bullet }\to x}$ and if for every ${\displaystyle l\in \mathbb {N} ,}$ ${\displaystyle x_{l}^{\bullet }\to x_{l},}$ then there exists a subsequence of the net ${\displaystyle \left(x_{l}^{i}\right)_{(l,i)\in \mathbb {N} \times \mathbb {N} }}$ that converges to ${\displaystyle x}$ (here, "subsequence" means a subnet that is also a sequence). Compare this statement to the diagonal principal used to define topologies in terms of nets. Importantly, there is in general no guarantee that the net ${\displaystyle \left(x_{l}^{i}\right)_{(l,i)\in \mathbb {N} \times \mathbb {N} }}$ will converge to ${\displaystyle x}$ (or even to any point at all).
4. Statement (3) but with the additional requirement that ${\displaystyle \iota >\lambda .}$

Properties[]

Every Fréchet–Urysohn space is a sequential space although the opposite implication is not true in general.^[2][3]

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

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 ${\displaystyle (X,\tau )}$ on a finite set ${\displaystyle X}$ is a Fréchet–Urysohn space.

Metrizable continuous dual spaces[]

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]

Sequential spaces that are not Fréchet–Urysohn[]

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

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 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]

See also[]

• Axioms of countability
• First-countable space – Topological space where each point has a countable neighbourhood basis
• Limit point – Point in a topological space
• Sequence covering maps
• Sequential space – Topological space with specific properties

Notes[]

Citations[]

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., 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.
• Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.