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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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