Fréchet–Urysohn space

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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

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Let   be a topological space. The sequential closure of   in   is the set:  

where   or   may be written if clarity is needed.

A topological space   is said to be a Fréchet–Urysohn space if  

for every subset   where   denotes the closure of   in  

Sequentially open/closed sets

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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  

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  

Strong Fréchet–Urysohn space

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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

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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

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If   is a topological space then the following are equivalent:

  1.   is a Fréchet–Urysohn space.
  2. Definition:   for every subset  
  3.   for every subset  
    • This statement is equivalent to the definition above because   always holds for every  
  4. Every subspace of   is a sequential space.
  5. 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

  then there exist strictly increasing maps   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

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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

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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

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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.[6]

Sequential spaces that are not Fréchet–Urysohn

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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   Give   its usual topology   in which a subset   is open (resp. closed) if and only if for every integer   the set   is an open (resp. closed) subset of   (with it usual Euclidean topology). If   and   is a sequence in   then   in   if and only if there exists some integer   such that both   and   are contained in   and   in   From these facts, it follows that   is a sequential space. For every integer   let   denote the open ball in   of radius   (in the Euclidean norm) centered at the origin. Let   Then the closure of   is   is all of   but the origin   of   does not belong to the sequential closure of   in   In fact, it can be shown that   This proves that   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

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  • 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.
  • 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

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  1. ^ 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.
  2. ^ 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

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  1. ^ Arkhangel'skii, A.V. and Pontryagin L.S.,  General Topology I, definition 9 p.12
  2. ^ Engelking 1989, Exercise 2.1.H(b)
  3. ^ Engelking 1989, Example 1.6.18
  4. ^ Ma, Dan (19 August 2010). "A note about the Arens' space". Retrieved 1 August 2013.
  5. ^ Gabriyelyan, S.S. "On topological spaces and topological groups with certain local countable networks (2014)
  6. ^ Trèves 2006, p. 201.
  7. ^ "Topological vector space". Encyclopedia of Mathematics. Encyclopedia of Mathematics. Retrieved September 6, 2020. It is a Montel space, hence paracompact, and so normal.
  8. ^ Gabriyelyan, Saak "Topological properties of Strict LF-spaces and strong duals of Montel Strict LF-spaces" (2017)
  9. ^ T. Shirai, Sur les Topologies des Espaces de L. Schwartz, Proc. Japan Acad. 35 (1959), 31-36.

References

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