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Wikipedia / Sequential space

In topology and related fields of mathematics, a sequential space is a topological space whose topology can be completely characterized by its convergent/divergent sequences. They can be thought of as spaces that satisfy a very weak axiom of countability, and all first-countable spaces (notably metric spaces) are sequential.

if a convergent sequence is contained in a closed set then the limit as well. Sets with this property are known as sequentially closed. Sequential spaces are precisely those topological spaces for which sequentially closed sets are in fact closed. (These definitions can also be rephrased in terms of sequentially open sets; see below.) Said differently, any topology can be described in terms of nets (also known as Moore–Smith sequences), but those sequences may be "too long" (indexed by too large an ordinal) to compress into a sequence. Sequential spaces are those topological spaces for which nets of countable length (i.e., sequences) suffice to describe the topology.

Any topology can be refined (that is, made finer) to a sequential topology, called the sequential coreflection

The related concepts of Fréchet–Urysohn spaces, -sequential spaces are also defined in terms of how a space's topology interacts with sequences, but have subtly different properties.

-sequential spaces were introduced by S. P. Franklin.

History

Although spaces satisfying such properties had implicitly been studied for several years, the first formal definition is due to S. P. Franklin in 1965. Franklin wanted to determine "the classes of topological spaces that can be specified completely by the knowledge of their convergent sequences", and began by investigating the first-countable spaces, for which it was already known that sequences sufficed. Franklin then arrived at the modern definition by abstracting the necessary properties of first-countable spaces.

Preliminary definitions

be a sequence, indexed by the natural numbers

be a topology converges

between topological spaces is sequentially continuous

Sequential closure/interior

be a subset. The topological closure (resp. topological interior).

The sequential closurewhich defines a map, the sequential closure operator but the reverse may fail.

The sequential interior(the topological space again indicated with a subscript if necessary).

Sequential closure and interior satisfy many of the nice properties of topological

; ; ; ; and

That is, sequential closure is a preclosure operator. Unlike topological closure, sequential closure is not idempotent: the last containment may be strict. Thus sequential closure is not a (Kuratowski) closure operator.

Sequentially closed and open sets

is defined to be sequentially open if its complement is sequentially closed. Equivalent conditions include:

or ).

is a sequential neighborhood in its sequential interior; sequential neighborhoods need not be sequentially open (see below).

to be sequentially open but not open. Similarly, it is possible for there to exist a sequentially closed subset that is not closed.

Sequential spaces and coreflection

As discussed above, sequential closure is not in general idempotent, and so not the closure operator of a topology. One can obtain an idempotent sequential closure via transfinite iteration: for a successor ordinaland, for a limit ordinal (the first uncountable ordinal). Conversely, the sequential order the above sequence will stabilize.

The transfinite sequential closure is idempotent and thus a closure operator. In particular, it defines a topology, the sequential coreflection. In the sequential coreflection, every sequentially-closed set is closed (and every sequentially-open set is open).

Sequential spaces

is sequential if it satisfies any of the following equivalent conditions: is its own sequential coreflection. is open. is closed. that is there exists some is continuous if and only if it is sequentially continuous). is the quotient of a first-countable space. is the quotient of a metric space.

).

T- and N-sequential spaces

A -sequential space is a topological space with sequential order 1, which is equivalent to any of the following conditions: is sequentially closed (resp. open). are idempotent. ; formally, sequentially-open neighborhoods are a neighborhood basis for the sequential neighborhoods.

Being a -sequential space is incomparable with being a sequential space; there are sequential spaces that are not is called a -sequential (or neighborhood-sequential) if it is both sequential and -sequential. An equivalent condition is that every sequential neighborhood contains an open (classical) neighborhood.

Every first-countable space (and thus every metrizable space-sequential. There exist topological vector spaces that are sequential but -sequential (and thus not -sequential).

Fréchet–Urysohn spaces

is called Fréchet–Urysohn if it satisfies any of the following equivalent conditions: is hereditarily sequential; that is, every topological subspace is sequential.

Fréchet–Urysohn spaces are also sometimes said to be "Fréchet," but should be confused with neither Fréchet spaces in functional analysis nor the T1 condition.

Examples and sufficient conditions

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.

and identify of integers to a point. As a quotient of a metric space, the result is sequential, but it is not first countable.

Every first-countable space is Fréchet–Urysohn and every Fréchet-Urysohn space is sequential. Thus every metrizable or pseudometrizable space in particular, every second-countable space, metric space, or discrete space is sequential.

be a set of maps from Fréchet–Urysohn Then the final topology is sequential.

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

Spaces that are sequential but not Fréchet-Urysohn

Schwartz space of smooth functions, as discussed in the article on distributions, are both widely-used sequential spaces.

More generally, every infinite-dimensional Montel DF-space is sequential but not Fréchet–Urysohn.

Arens' space is sequential, but not Fréchet–Urysohn.

Non-examples (spaces that are not sequential)

The simplest 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. (One could call the topology "sequentially discrete".)

denote the space of-smooth test functions denote the space of distributions, the strong dual space; neither are sequential (nor even an Ascoli space). are Montel spaces and, 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 (that is, converges pointwise).

Consequences

Every sequential space has countable tightness and is compactly generated.

is a continuous open surjection is injective.)

bases is an open map that is eventually in

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 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 (that is, 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".

Every sequential space is compactly generated, and finite products in Seq coincide with those for compactly generated spaces, since products in the category of compactly generated spaces preserve quotients of metric spaces.

See also

Notes

Citations

References