Sequential space

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In topology and related fields of mathematics, a sequential space is a topological space that satisfies a very weak axiom of countability.

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In any topological space every open subset has the following property: if a sequence in converges to some point in then the sequence will eventually be entirely in (i.e. there exists an integer such that all belong to ); Any set with this property is said to be sequentially open, regardless of whether or not it is open in However, it is possible for there to exist a subset that has this property but fails to be an open subset of Sequential spaces are exactly those topological spaces where a subset with this property never fails to be open. Sequential spaces can be viewed as exactly those spaces where for any single given subset knowledge of which sequences in converge to which point(s) of (and which don't) is sufficient to determine whether or not is closed 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]

Alternatively, a space being sequential means that its topology if "forgotten", can be completely reconstructed using only sequences if one has all possible information about the convergence (or non-convergence) of sequences in and nothing more. However, like all topologies, any topology that cannot be described entirely in terms of sequences can nevertheless be described entirely in terms of nets (also known as Moore–Smith sequences) or alternatively, in terms of filters. All first-countable spaces, which includes metric spaces, are sequential spaces.

There are other classes of topological spaces, such as Fréchet–Urysohn spaces, T-sequential spaces, and -sequential spaces, that are also defined in terms of how a space's topology interacts with sequences. Their definitions differ from that of sequential spaces in only subtle (but important) ways and it is often (initially) surprising that a sequential space does not necessarily have the properties of a Fréchet–Urysohn, T-sequential, or -sequential space.

Sequential spaces and -sequential spaces were introduced by S. P. Franklin. [1]

History

Although spaces satisfying such properties had implicitly been studied for several years, the first formal definition is originally due to S. P. Franklin in 1965, who was investigating the question of "what are the classes of topological spaces that can be specified completely by the knowledge of their convergent sequences?" Franklin arrived at the definition above by noting that every first-countable space can be specified completely by the knowledge of its convergent sequences, and then he abstracted properties of first countable spaces that allowed this to be true.

Definitions

Preliminaries

Let be a set and let be a sequence in where a sequence in a set is by definition just a map from the natural numbers into If is a set then means that the is a sequence in If is a map then denotes the sequence Because the sequence is just a function this is consistent with the definition of function composition, meaning that

For any index the tail of starting at is the set:

The set of all tails of is denoted by

and it forms a filter base (also called a prefilter) on which is why it is called the prefilter of tails or the sequential filter base of tails of

If is a subset then a sequence in is eventually in if there exists some index such that (that is, for any integer such that ).

Let be a topological space (not necessarily Hausdorff) and let be a sequence in The sequence converges in to a point written in and is called a limit point of if for every neighborhood of in is eventually in As usual, the notation mean that in and is the only limit point of in that is, if in then If is not Hausdorff then it is possible for a sequence to converge to two or more distinct points.

A point is called a cluster point or accumulation point of in if for every neighborhood of in and every there exists some integer such that (or said differently, if and only if for every neighborhood of and every ).

Sequential closure/interior

Let be a topological space and let be a subset. The topological closure (resp. topological interior) of in is denoted by (resp. ).

The sequential closure of in is the set:

where or may be written if clarity is needed. The inclusion always holds but in general, set equality may fail to hold. The sequential closure operator is the map defined by where denotes the power set of

The sequential interior of in is the set:

where or may be written if clarity is needed.

As with the topological closure operator, and always hold and for all subsets

so that consequently,

However, it is in general possible that which in particular would imply that because the topological closure operator is idempotent, meaning that for all subsets

Transfinite sequential closure

The transfinite sequential closure is defined as follows: define to be define to be and for a limit ordinal define to be Then there is a smallest ordinal such that and for this is called the transfinite sequential closure of In fact, always holds where is the first uncountable ordinal. The transfinite sequential closure of is sequentially closed. Taking the transfinite sequential closure solves the idempotency problem above. The smallest such that for each is called sequential order of the space [2] This ordinal invariant is well-defined for sequential spaces.

Sequentially open/closed sets

Let be a topological space (not necessarily Hausdorff) and let be a subset. It is known that the subset is open in if and only if whenever is a net in that converges in to a point then is eventually in where "eventually in " means that there exists some index such that for all satisfying The definition of a sequentially open subset of uses a variation of this characterization in which nets are replaced with sequences.

The set is called sequentially open if it satisfies any of the following equivalent conditions:

  1. Definition: Whenever a sequence in converges to some point of then that sequence is eventually in
  2. If is a sequence in and if there exists some is such that in then is eventually in (that is, there exists some integer such that the tail ).
  3. The set is sequentially closed in

The set is called sequentially closed if it satisfies any of the following equivalent conditions:

  1. Definition: Whenever a sequence in converges in to some point then
  2. If is a sequence in and if there exists some is such that in then
  3. The set is sequentially open in

The complement of a sequentially open set is a sequentially closed set, and vice versa.

The set is called a sequential neighborhood of a point if it satisfies any of the following equivalent conditions:

  1. Definition:
    • Importantly, " is a sequential neighborhood of " is not defined as: "there exists a sequentially open set such that "
  2. Any sequence in that converges to is eventually in

Let

denote the set of all sequentially open subsets of where this may be denoted by is the topology is understood. Every open (resp. closed) subset of is sequentially open (resp. sequentially closed), which implies that

It is possible for the containment to be proper, meaning that there may exist a subset of that is sequentially open but not open. Similarly, it is possible for there to exist a sequentially closed subset that is not closed.

Sequential spaces

A topological space is called a sequential space if it satisfies any of the following equivalent conditions:

  1. Definition: Every sequentially open subset of is open.
  2. Every sequentially closed subset of is closed.
  3. For any subset that is not closed in there exists some for which there exists a sequence in that converges to [3]
    For any subset that is not closed in and for every there exists a sequence in that converges to
    • This makes it obvious that every Fréchet–Urysohn space is a sequential space.
  4. is the quotient of a first countable space.
  5. is the quotient of a metric space.
  6. Universal property of sequential spaces: For every topological space a map is continuous if and only if it is sequentially continuous.
    • A map is called sequentially continuous if for every and every sequence in if in then in This condition is equivalent to the map being continuous.
    • Every continuous map is necessarily sequentially continuous but in general, the converse may fail to hold.

By taking and to be the identity map on in the last condition, it follows that the class of sequential spaces consists precisely of those spaces whose topological structure is determined by convergent sequences.

Proof of the equivalences

(1) (2): Assume that any sequentially open subsets is open and let be sequentially closed. It is proved above that the complement is sequentially open and thus open so that is closed. The converse is similar.

(2) (3): Contraposition of 2 says that " not closed implies not sequentially closed", and hence there exists a sequence of elements of that converges to a point outside of Since the limit is necessarily adherent to it is in the closure of

Conversely, suppose for a contradiction that a subset is sequentially closed but not closed. By 3, there exists a sequence in that converges to a point in i.e. the limit lies outside This contradicts sequential closedness of

T-sequential and -sequential spaces

A sequential space may fail to be a T-sequential space and also a T-sequential space may fail to be a sequential space. In particular, it should not be assumed that a sequential space has the properties described in the next definitions.

A topological space is called a T-sequential space (or topological-sequential) if it satisfies one of the following equivalent conditions: [1]

  1. Definition: The sequential interior of every subset of is sequentially open.
  2. The sequential closure of every subset of is sequentially closed.
  3. For all
    • The inclusion always holds for every
  4. For all
    • The inclusion always holds for all
  5. For all is equal to the union of all subsets of that are sequentially open in
  6. For all is equal to the intersection of all subsets of that contain and are sequentially closed in
  7. For all the collection of all sequentially open neighborhoods of in forms a neighborhood basis at for the set of all sequential neighborhoods of
    • This means for any and any sequential neighborhood of there exists a sequentially open set such that
    • Here, the exact definition of "sequential neighborhood" is important because recall that " is a sequential neighborhood of " was defined to mean that
  8. For any and any sequential neighborhood of there exists a sequential neighborhood of such that for every the set is a sequential neighborhood of

As with T-sequential spaces, it should not be assumed that a sequential space has the properties described in the next definition.

A topological space is called an -sequential (or neighborhood-sequential) space if it satisfies any of the following equivalent conditions: [1]

  1. Definition: For every if a set is a sequential neighborhood of then is a neighborhood of in
    • Recall that being a sequential neighborhood (resp. a neighborhood) of means that (resp. ).
  2. is both sequential and T-sequential.

Every first-countable space is -sequential. [1] There exist topological vector spaces that are sequential but not-sequential (and thus not T-sequential). [1] where recall that every metrizable space is first countable. There also exist topological vector spaces that are T-sequential but not sequential. [1]

Fréchet–Urysohn spaces

Every Fréchet–Urysohn space is a sequential space but there exist sequential spaces that are not Fréchet–Urysohn. [4] [5] Consequently, it should not be assumed that a sequential space has the properties described in the next definition.

A topological space is called Fréchet–Urysohn space if it satisfies any of the following equivalent conditions:

  1. Definition: For every subset
  2. Every topological subspace of is a sequential space.
  3. For any subset that is not closed in and for every there exists a sequence in that converges to

Fréchet–Urysohn spaces are also sometimes said to be Fréchet, which should not be confused with Fréchet spaces in functional analysis; confusingly, Fréchet space in topology is also sometimes used as a synonym for T1 space.

Topology of sequentially open sets

Let denote the set of all sequentially open subsets of the topological space Then is a topology on that contains the original topology that is,

Proofs

Let be sequentially open. It is now shown that its complement is sequentially closed; that is, that a convergent sequence of elements of has its limit in Suppose for a contradiction that then there exists some integer such that which contradicts the fact that all are supposed to be in

The converse will now be shown; that is, it is now shown that if is sequentially closed then its complement is sequentially open. Let be a sequence in such that and suppose for a contradiction that for any i.e. for all integers there exists Define by recursion the subsequence of elements of  : set and then that is, and It is convergent as a subsequence of a convergent sequence, and all its elements are in Hence the limit has to be in which contradicts that The sequence is therefore eventually in

It is now shown that the set of sequentially open subsets is a topology. Specifically, this means that and are sequentially open, arbitrary unions of sequentially open subset is sequentially open and finite intersections of sequentially open subsets is sequentially open. Any empty sequence satisfies any property and any sequence in is eventually in Let be a family of sequentially open subsets, let and let be a sequence in converging to being in the union means there exists such that and by sequential openness, the sequence is eventually in Finally, if is a finite intersection of sequentially open subsets, then a sequence converging to eventually converges to each of the i.e. for all satisfying there exists some such that Taking one has

The generated sequential topology is finer than the original one, meaning that if is open, then it is sequentially open. Let be a sequence in converging to Since is open, it is a neighborhood of and by definition of convergence, there exists such that

The topological space is said to be sequentially Hausdorff if is a Hausdorff space.

Properties of the topology of sequentially open sets

Every sequential space has countable tightness.

The topological space is always a sequential space (even if is not), [6] and has the same convergent sequences and limits as Explicitly, this means that if and is a sequence in then in if and only if in

If is any topology on such that for every and every sequence in

then necessarily

If is continuous then so is

Sequential continuity

A map is called sequentially continuous if for every sequence in and every if in then necessarily in which happens if and only if

is continuous.

Every continuous map is sequentially continuous although in general, the converse may fail to hold. In fact, a space is a sequential space if and only if it has the following universal property for sequential spaces:

For every topological space and every map the map is continuous if and only if it is sequentially continuous.

Sufficient conditions

Every first-countable space is sequential, hence each second-countable space, metric space, and discrete space is sequential. Every first-countable space is a Fréchet–Urysohn space and every Fréchet-Urysohn space is sequential. Thus every metrizable and pseudometrizable space is a sequential space and a Fréchet–Urysohn space.

A Hausdorff topological vector space is sequential if and only if there exists no strictly finer topology with the same convergent sequences. [7] [8]

Let be a set and let be a family of -valued maps with each map being of the form where the domain is some topological space. If every domain is a Fréchet–Urysohn space then the final topology on induced by makes into a sequential space.

Examples

Every CW-complex is sequential, as it can be considered as a quotient of a metric space. The prime spectrum of a commutative Noetherian ring with the Zariski topology is sequential.

Sequential spaces that are not first countable

Take the real line and identify the set of integers to a point. It is a sequential space since it is a quotient of a metric space. But it is not first countable.

Sequential spaces that are not Fréchet–Urysohn spaces

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 [9] 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. [10] [11]

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

Examples of non-sequential spaces

Spaces of test functions and distributions

Let denote the space of test functions with its canonical LF topology, which makes it into a distinguished strict LF-space and let denote the space of distributions, which by definition is the strong dual space of These two space, which completely underpin the theory of distributions and which have many nice properties, are nevertheless prominent examples of spaces that are not sequential spaces (and thus neither Fréchet–Urysohn spaces nor -sequential spaces).

Both and are complete nuclear Montel ultrabornological spaces, which implies that all four of these locally convex spaces are also paracompact [9] normal reflexive barrelled spaces. It is known that in the dual space of any Montel space, a sequence of continuous linear functionals converges in the strong dual topology if and only if it converges in the weak* topology (i.e. pointwise), [12] which in particular, is the reason why a sequence of distributions converges in (with is given strong dual topology) if and only if it converges pointwise. The space is also a Schwartz topological vector space. Nevertheless, neither nor its strong dual is a sequential space (not even an Ascoli space). [10] [11]

Cocountable topology

Another example of a space that is not sequential is the cocountable topology on an uncountable set. Every convergent sequence in such a space is eventually constant, hence every set is sequentially open. But the cocountable topology is not discrete. In fact, one could say that the cocountable topology on an uncountable set is "sequentially discrete".

Properties

If is a continuous open surjection between two Hausdorff sequential spaces then the set is a closed subset of the set is a closed subset of that satisfies and the restriction is injective.

If is a surjective map (not assumed to be continuous) onto a Hausdorff sequential space and if is a basis for the topology on then is an open map if and only if for every and every basic neighborhood of if in then necessarily Here, denotes the image (or range) of the sequence/map

Categorical properties

The full subcategory Seq of all sequential spaces is closed under the following operations in the category Top of topological spaces:

The category Seq is not closed under the following operations in Top:

Since they are closed under topological sums and quotients, the sequential spaces form a coreflective subcategory of the category of topological spaces. In fact, they are the coreflective hull of metrizable spaces (i.e., the smallest class of topological spaces closed under sums and quotients and containing the metrizable spaces).

The subcategory Seq is a Cartesian closed category with respect to its own product (not that of Top). The exponential objects are equipped with the (convergent sequence)-open topology. P.I. Booth and A. Tillotson have shown that Seq is the smallest Cartesian closed subcategory of Top containing the underlying topological spaces of all metric spaces, CW-complexes, and differentiable manifolds and that is closed under colimits, quotients, and other "certain reasonable identities" that Norman Steenrod described as "convenient".

See also

Notes

  1. This interpretation assumes that you make this determination only to the given set and not to other sets; said differently, you cannot simultaneously apply this "test" to infinitely many subsets (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 any set other than There exist sequential spaces that are not 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

  1. 1 2 3 4 5 6 Snipes, Ray F. "T-sequential topological spaces"
    • Arhangel'skiĭ, A. V.; Franklin, S. P. (1968). "Ordinal invariants for topological spaces". Michigan Math. J. 15 (3): 313–320. doi: 10.1307/mmj/1029000034 .
  2. Arkhangel'skii, A.V. and Pontryagin L.S.,  General Topology I, definition 9 p.12
  3. Engelking 1989, Example 1.6.18
  4. Ma, Dan (19 August 2010). "A note about the Arens' space" . Retrieved 1 August 2013.
  5. "Topology of sequentially open sets is sequential?".
  6. Wilansky 2013, p. 224.
  7. Dudley, R. M., On sequential convergence - Transactions of the American Mathematical Society Vol 112, 1964, pp. 483-507
  8. 1 2 "Topological vector space". Encyclopedia of Mathematics. Encyclopedia of Mathematics. Retrieved September 6, 2020. It is a Montel space, hence paracompact, and so normal.
  9. 1 2 Gabriyelyan, Saak "Topological properties of Strict LF-spaces and strong duals of Montel Strict LF-spaces" (2017)
  10. 1 2 T. Shirai, Sur les Topologies des Espaces de L. Schwartz, Proc. Japan Acad. 35 (1959), 31-36.
  11. Trèves 2006, pp. 351-359.

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