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 are sequential.
In any topological space if a convergent sequence is contained in a closed set then the limit of that sequence must be contained in 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. Said differently, any topology can be described in terms of nets, but those sequences may be "too long" to compress into a sequence. Sequential spaces are those topological spaces for which nets of countable length suffice to describe the topology.
Any topology can be refined to a sequential topology, called the sequential coreflection of
The related concepts of Fréchet–Urysohn spaces, -sequential spaces, and -sequential spaces are also defined in terms of how a space's topology interacts with sequences, but have subtly different properties.
Sequential spaces and -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

Let be a set and let be a sequence in ; that is, a family of elements of, indexed by the natural numbers. In this article, means that each element in the sequence is an element of and, if is a map, then For any index the tail of starting at is the sequence A sequence is eventually in if some tail of satisfies
Let be a topology on and a sequence therein. The sequence converges to a point written , if, for every neighborhood of eventually is in is then called a limit point of
A function between topological spaces is sequentially continuous if implies

Sequential closure/interior

Let be a topological space and let be a subset. The topological closure of in is denoted by .
The sequential closure of in is the setwhich defines a map, the sequential closure operator, on the power set of If necessary for clarity, this set may also be written or It is always the case that but the reverse may fail.
The sequential interior of in is the set.
Sequential closure and interior satisfy many of the nice properties of topological closure and interior: for all subsets

and ;

Proof

Fix If then there exists with But by the definition of sequential interior, eventually is in contradicting

Conversely, suppose ; then there exists a sequence with that is not eventually in By passing to the subsequence of elements not in we may assume that But then

and ;
;
; 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 closure operator.

Sequentially closed and open sets

A set is sequentially closed if ; equivalently, for all and such that we must have
A set is defined to be sequentially open if its complement is sequentially closed. Equivalent conditions include:

or
For all and such that eventually is in .

A set is a sequential neighborhood of a point if it contains in its sequential interior; sequential neighborhoods need not be sequentially open.
It is possible for a subset of 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 ordinal define and, for a limit ordinal defineThis process gives an ordinal-indexed increasing sequence of sets; as it turns out, that sequence always stabilizes by index . Conversely, the sequential order of is the minimal ordinal at which, for any choice of the above sequence will stabilize.
The transfinite sequential closure of is the terminal set in the above sequence: The operator 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.

Sequential spaces

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

is its own sequential coreflection.
Every sequentially open subset of is open.
Every sequentially closed subset of is closed.
For any subset that is closed in there exists some and a sequence in that converges to
For every topological space a map 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.

By taking and to be the identity map on in the universal property, it follows that the class of sequential spaces consists precisely of those spaces whose topological structure is determined by convergent sequences. If two topologies agree on convergent sequences, then they necessarily have the same sequential coreflection. Moreover, a function from is sequentially continuous if and only if it is continuous on the sequential coreflection.

- and -sequential spaces

A -sequential space is a topological space with sequential order 1, which is equivalent to any of the following conditions:

The sequential closure of every subset of is sequentially closed.
or are idempotent.
or
Any sequential neighborhood of can be shrunk to a sequentially-open set that contains ; formally, sequentially-open neighborhoods are a neighborhood basis for the sequential neighborhoods.
For any and any sequential neighborhood of there exists a sequential neighborhood of such that, for every the set is a sequential neighborhood of

Being a -sequential space is incomparable with being a sequential space; there are sequential spaces that are not -sequential and vice-versa. However, a topological space is called a -sequential if it is both sequential and -sequential. An equivalent condition is that every sequential neighborhood contains an open neighborhood.
Every first-countable space is -sequential. There exist topological vector spaces that are sequential but -sequential.

Fréchet–Urysohn spaces

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

is hereditarily sequential; that is, every topological subspace is sequential.
For every subset
For any subset that is not closed in and every there exists a sequence in that converges to

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.
Take the real line and identify the set 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.
Let be a set of maps from Fréchet–Urysohn spaces to Then the final topology that induces on 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

and the 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.
Let denote the space of -smooth test functions with its canonical topology and let denote the space of distributions, the strong dual space of ; neither are sequential. On the other hand, both and 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.

Consequences

Every sequential space has countable tightness and is compactly generated.
If is a continuous open surjection between two Hausdorff sequential spaces then the set of points with unique preimage is closed.
If is a surjective map onto a Hausdorff sequential space and bases for the topology on then is an open map if and only if, for every basic neighborhood of and sequence in there is a subsequence of that is eventually in