Fréchet–Urysohn space
Property of topological space
(Learn how and when to remove this message) In the field of topology, a Fréchet–Urysohn space is a topological space
with the property that for every subset
the closure of
in
is identical to the sequential closure of
in
Fréchet–Urysohn spaces are a special type of sequential space.
The property is named after Maurice Fréchet and Pavel Urysohn.
Definitions
Let
be a topological space. The sequential closure of
in
is the set:
![{\displaystyle {\begin{alignedat}{4}\operatorname {scl} S:&=[S]_{\operatorname {seq} }:=\left\{x\in X~:~{\text{ there exists a sequence }}s_{\bullet }=\left(s_{i}\right)_{i=1}^{\infty }\subseteq S{\text{ in }}S{\text{ such that }}s_{\bullet }\to x{\text{ in }}(X,\tau )\right\}\end{alignedat}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4a3e0b67986eba190abb38e7ace8ca4282a2be0b)
where
or
may be written if clarity is needed.
A topological space
is said to be a Fréchet–Urysohn space if
![{\displaystyle \operatorname {cl} _{X}S=\operatorname {scl} _{X}S}](https://wikimedia.org/api/rest_v1/media/math/render/svg/66fd805b5cd8bcc8748f7ff86d8b9bb9bd63d88a)
for every subset
where
denotes the closure of
in
Sequentially open/closed sets
Suppose that
is any subset of
A sequence
is eventually in
if there exists a positive integer
such that
for all indices
The set
is called sequentially open if every sequence
in
that converges to a point of
is eventually in
; Typically, if
is understood then
is written in place of
The set
is called sequentially closed if
or equivalently, if whenever
is a sequence in
converging to
then
must also be in
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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/84b7adfc045ddb8f749199bc99504cfb68552627)
denote the set of all sequentially open subsets of
where this may be denoted by
is the topology
is understood. The set
is a topology on
that is finer than the original topology
Every open (resp. closed) subset of
is sequentially open (resp. sequentially closed), which implies that
![{\displaystyle \tau \subseteq \operatorname {SeqOpen} (X,\tau ).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3fb4a59b00484c153aa76912228c2679fe7f1700)
Strong Fréchet–Urysohn space
A topological space
is a strong Fréchet–Urysohn space if for every point
and every sequence
of subsets of the space
such that
there exist a sequence
in
such that
for every
and
in
The above properties can be expressed as selection principles.
Contrast to sequential spaces
Every open subset of
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
where for any single given subset
knowledge of which sequences in
converge to which point(s) of
(and which do not) is sufficient to determine whether or not
is closed in
(respectively, is sufficient to determine the closure of
in
).[note 1] Thus sequential spaces are those spaces
for which sequences in
can be used as a "test" to determine whether or not any given subset is open (or equivalently, closed) in
; 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
is a topological space then the following are equivalent:
is a Fréchet–Urysohn space. - Definition:
for every subset ![{\displaystyle S\subseteq X.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c586f8a77721c22346403a73aa1233c9cebd4f4d)
for every subset
- This statement is equivalent to the definition above because
always holds for every ![{\displaystyle S\subseteq X.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c586f8a77721c22346403a73aa1233c9cebd4f4d)
- Every subspace of
is a sequential space. - For any subset
that is not closed in
and for every
there exists a sequence in
that converges to
- Contrast this condition to the following characterization of a sequential space:
- For any subset
that is not closed in
there exists some
for which there exists a sequence in
that converges to
[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. In the following characterization, all convergence is assumed to take place in
If
is a Hausdorff sequential space then
is a Fréchet–Urysohn space if and only if the following condition holds: If
is a sequence in
that converge to some
and if for every
is a sequence in
that converges to
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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/698d2766eaf681ad82ab7708a9a69fd6828c4e97)
then there exist strictly increasing maps
![{\displaystyle \iota ,\lambda :\mathbb {N} \to \mathbb {N} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/88a079029a39d568cc32f7ca2359d8c699951456)
such that
(It suffices to consider only sequences
with infinite ranges (i.e.
is infinite) because if it is finite then Hausdorffness implies that it is necessarily eventually constant with value
in which case the existence of the maps
with the desired properties is readily verified for this special case (even if
is not a Fréchet–Urysohn space).
Properties
Every subspace of a Fréchet–Urysohn space is Fréchet–Urysohn.[2]
Every Fréchet–Urysohn space is a sequential space although the opposite implication is not true in general.[3][4]
If a Hausdorff locally convex topological vector space
is a Fréchet-Urysohn space then
is equal to the final topology on
induced by the set
of all arcs in
which by definition are continuous paths
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
on a finite set
is a Fréchet–Urysohn space.
Metrizable continuous dual spaces
A metrizable locally convex topological vector space (TVS)
(for example, a Fréchet space) is a normable space if and only if its strong dual space
is a Fréchet–Urysohn space,[5] or equivalently, if and only if
is a normable space.
Sequential spaces that are not Fréchet–Urysohn
Direct limit of finite-dimensional Euclidean spaces
The space of finite real sequences
is a Hausdorff sequential space that is not Fréchet–Urysohn. For every integer
identify
with the set
where the latter is a subset of the space of sequences of real numbers
explicitly, the elements
and
are identified together. In particular,
can be identified as a subset of
and more generally, as a subset
for any integer
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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a1e5380bdba54849f86f9bcc2a7c174f752f6ec7)
Give
![{\displaystyle \mathbb {R} ^{\infty }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/da5e5160fa2811da2c516b0fa543236c5cf707fc)
its usual topology
![{\displaystyle \tau ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/26d6cc28c28ff4ff88402f47f2a99e583e9e045f)
in which a subset
![{\displaystyle S\subseteq \mathbb {R} ^{\infty }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/abe25f75a08b57d8b1aab855c9fc625a6138ee7c)
is open (resp. closed) if and only if for every integer
![{\displaystyle n\geq 1,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cc38ec6af7dd11fdc9baa67365f23906d76da4bb)
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\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/efcfad181e889c869cecf58e3adea3a0fa1e609c)
is an open (resp. closed) subset of
![{\displaystyle \mathbb {R} ^{n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d)
(with it usual
Euclidean topology). If
![{\displaystyle v\in \mathbb {R} ^{\infty }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bf46524b20e6fe9fb98874ee8ff3cef16cf36344)
and
![{\displaystyle v_{\bullet }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/37c6a30c7ee6bfef9df6120bbb1d0e636c49b454)
is a sequence in
![{\displaystyle \mathbb {R} ^{\infty }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/da5e5160fa2811da2c516b0fa543236c5cf707fc)
then
![{\displaystyle v_{\bullet }\to v}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5c69fa41987d0f6921e6fd8dcca08f076902b9c2)
in
![{\displaystyle \left(\mathbb {R} ^{\infty },\tau \right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c5dfac3f1c596751f7ebfd107bf2685c2db37e71)
if and only if there exists some integer
![{\displaystyle n\geq 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d8ce9ce38d06f6bf5a3fe063118c09c2b6202bfe)
such that both
![{\displaystyle v}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597)
and
![{\displaystyle v_{\bullet }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/37c6a30c7ee6bfef9df6120bbb1d0e636c49b454)
are contained in
![{\displaystyle \mathbb {R} ^{n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d)
and
![{\displaystyle v_{\bullet }\to v}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5c69fa41987d0f6921e6fd8dcca08f076902b9c2)
in
![{\displaystyle \mathbb {R} ^{n}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/76ef548febfc9981762740107858be9e3a5576c3)
From these facts, it follows that
![{\displaystyle \left(\mathbb {R} ^{\infty },\tau \right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c5dfac3f1c596751f7ebfd107bf2685c2db37e71)
is a sequential space. For every integer
![{\displaystyle n\geq 1,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cc38ec6af7dd11fdc9baa67365f23906d76da4bb)
let
![{\displaystyle B_{n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2f568bf6d34e97b9fdda0dc7e276d6c4501d2045)
denote the open ball in
![{\displaystyle \mathbb {R} ^{n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d)
of radius
![{\displaystyle 1/n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f0e10667bad240500f5044257143510127e03d69)
(in the
Euclidean norm) centered at the origin. Let
![{\displaystyle S:=\mathbb {R} ^{\infty }\,\setminus \,\bigcup _{n=1}^{\infty }B_{n}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5bb88f7268ebd378859f87fd5e4707648244fb19)
Then the closure of
![{\displaystyle S}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2)
is
![{\displaystyle \left(\mathbb {R} ^{\infty },\tau \right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c5dfac3f1c596751f7ebfd107bf2685c2db37e71)
is all of
![{\displaystyle \mathbb {R} ^{\infty }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/da5e5160fa2811da2c516b0fa543236c5cf707fc)
but the origin
![{\displaystyle (0,0,0,\ldots )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cf8f05f7a603ace331accbac5bcfb60b6c6020e7)
of
![{\displaystyle \mathbb {R} ^{\infty }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/da5e5160fa2811da2c516b0fa543236c5cf707fc)
does
not belong to the sequential closure of
![{\displaystyle S}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2)
in
![{\displaystyle \left(\mathbb {R} ^{\infty },\tau \right).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6a5080d0ba62f78e341386dee052df13ebb78b17)
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 )\}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f08363df10b4f848a0a2a7c1ff1daec841813f7c)
This proves that
![{\displaystyle \left(\mathbb {R} ^{\infty },\tau \right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c5dfac3f1c596751f7ebfd107bf2685c2db37e71)
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
and the space of smooth functions
The following extensively used spaces are prominent examples of sequential spaces that are not Fréchet–Urysohn spaces. Let
denote the Schwartz space and let
denote the space of smooth functions on an open subset
where both of these spaces have their usual Fréchet space topologies, as defined in the article about distributions. Both
and
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[7] normal reflexive barrelled spaces. The strong dual spaces of both
and
are sequential spaces but neither one of these duals is a Fréchet-Urysohn space.[8][9]
See also
- Axiom of countability – property of certain mathematical objects (usually in a category) that asserts the existence of a countable set with certain properties. Without such an axiom, such a set might not probably exist.Pages displaying wikidata descriptions as a fallback
- First-countable space – Topological space where each point has a countable neighbourhood basis
- Limit of a sequence – Value to which tends an infinite sequence
- Sequence covering map
- Sequential space – Topological space characterized by sequences
Notes
- ^ Of course, if you can determine all of the supersets of
that are closed in
then you can determine the closure of
So this interpretation assumes that you can only determine whether or not
is closed (and that this is 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
can be determined without it ever being necessary to consider a subset of
other than
this is not always possible in non-Fréchet-Urysohn spaces. - ^ 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
is necessarily sequentially open (resp. sequentially closed) so this "test" will never indicate "false" for any set
that really is open (resp. closed).
Citations
- ^ Arkhangel'skii, A.V. and Pontryagin L.S., General Topology I, definition 9 p.12
- ^ Engelking 1989, Exercise 2.1.H(b)
- ^ Engelking 1989, Example 1.6.18
- ^ Ma, Dan (19 August 2010). "A note about the Arens' space". Retrieved 1 August 2013.
- ^ Gabriyelyan, S.S. "On topological spaces and topological groups with certain local countable networks (2014)
- ^ "Topological vector space". Encyclopedia of Mathematics. Encyclopedia of Mathematics. Retrieved September 6, 2020.
It is a Montel space, hence paracompact, and so normal.
- ^ Gabriyelyan, Saak "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.
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.