Sequential space


In topology and related fields of mathematics, a sequential space is a topological space that satisfies a very weak axiom of countability. Sequential spaces are the most general class of spaces for which sequences suffice to determine the topology.
Every sequential space has countable tightness.

Definitions

Let X be a topological space and let S be a subset of X'.
The complement of a sequentially open set is a sequentially closed set, and vice versa.
Every open subset of X is sequentially open and every closed set is sequentially closed.
The converses are not generally true.
A sequential space is a space X satisfying one of the following equivalent conditions:
  1. Every sequentially open subset of X is open.
  2. Every sequentially closed subset of X is closed.
  3. For any subset S ⊆ X that is not closed, i.e. such that, there exists a sequence of elements of S that converges to an element of.
  4. X is the quotient of a first countable space.
  5. X is the quotient of a metric space.
  6. For every topological space Y and every map f : X → Y, we have that f is continuous if and only if for every sequence of points in X converging to x, we have converging to f''.
The final equivalent condition shows that the class of sequential spaces consists precisely of those spaces whose topological structure is determined by convergent sequences in the space.
Also, a space being sequential means that the original topology can be reconstructed by knowing the convergent sequences.

Sequential closure

Given a subset of a space, the sequential closure is the set
that is, the set of all points for which there is a sequence in that converges to. The map
is called the sequential closure operator. It shares some properties with ordinary closure, in that the empty set is sequentially closed:
Every closed set is sequentially closed:
for all ; here denotes the ordinary closure of the set. Sequential closure commutes with union:
for all. However, unlike ordinary closure, the sequential closure operator is not in general idempotent; that is, one may have that
and consequently, even when is a subset of a sequential space.
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. Taking the transfinite sequential closure solves the idempotency problem above.
The smallest such that for each is called sequential order of the space X. This ordinal invariant is well-defined for sequential spaces.

Topology of sequentially open sets

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

Properties of the topology of sequentially open sets

Topological spaces for which sequential closure is the same as ordinary closure are known as Fréchet–Urysohn spaces. That is, a Fréchet–Urysohn space has
for all. A space is a Fréchet–Urysohn space if and only if every subspace is a sequential space. Every first-countable space is a Fréchet–Urysohn space.
The space is named after Maurice Fréchet and Pavel Urysohn.
Clearly, every Fréchet–Urysohn space is a sequential space. The opposite implication is not true in general.
A topological space is a strong Fréchet–Urysohn space if for every point and every sequence of subsets of the space such that , there are points such that.
The above properties can be expressed as selection principles.

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.

Examples

Every first-countable space is sequential, hence each second-countable space, metric space, and discrete space is sequential.
Every Fréchet-Urysohn space is sequential.
Further examples are furnished by applying the categorical properties listed below. For example, every CW-complex is sequential, as it can be considered as a quotient of a metric space.
There are sequential spaces that are not first countable.
An 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".
The prime spectrum of a commutative Noetherian ring with the Zariski topology is sequential.

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.
The subcategory Seq is a Cartesian closed category with respect to its own product. The exponential objects are equipped with the -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".