Sequential space

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

Definitions

• Let X be a topological space. A subset U of X is sequentially open if each sequence (x_n) in X converging to a point of U is eventually in U (i.e. there exists N such that x_n is in U for all n ≥ N.)
• A subset F is sequentially closed if, whenever (x_n) is a sequence in F converging to x, then x must also be in F.

Every open subset of X is sequentially open.

A sequential space is a space 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.

Sequential closure

Given a subset $A\subset X$ of a space $X$, the sequential closure $[A]_{\mbox{seq}}$ is the set

$[A]_{\mbox{seq}}=\{x\in X:\{a_{n}\}\to x,a_{n}\in A\}$

that is, the set of all points $x\in X$ for which there is a sequence in $A$ that converges to $x$. The map

$[\;]_{\mbox{seq}}:A\mapsto [A]_{\mbox{seq}}$

is called the sequential closure operator. It shares some properties with ordinary closure, in that the empty set is sequentially closed:

$[\varnothing ]_{\mbox{seq}}=\varnothing$

Sequentially closed sets are subsets of closed sets:

$A\subset [A]_{\mbox{seq}}\subset {\overline {A}}$

for all $A\subset X$; here ${\overline {A}}$ denotes the ordinary closure of the set $A$. Sequential closure commutes with union:

$[A\cup B]_{\mbox{seq}}=[A]_{\mbox{seq}}\cup [B]_{\mbox{seq}}$

for all $A,B\subset X$. However, unlike ordinary closure, the sequential closure operator is not in general idempotent; that is, one may have that

$[[A]_{\mbox{seq}}]_{\mbox{seq}}\neq [A]_{\mbox{seq}}$

even when $A\subset X$ is a subset of a sequential space $X$.

Fréchet-Urysohn space

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

$[A]_{\mbox{seq}}={\overline {A}}$

for all $A\subset X$. A space is Fréchet-Urysohn 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.

History

Although spaces satisfying such properties had implicitly been studied for several years, the first formal definintion is originally due to S. P. Franklin in 1965, who was investigating the question of "what are the classes of topological spaces which 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 which allowed this to be true.

Examples

Every first countable space is sequential, hence each second countable, metric space, and discrete space is sequential. Further examples are furnished by applying the categorical properties listed below.

There are sequential spaces which are not first countable. (One example is to take the real line R and identify the set Z of integers to a point.)

An example of a space which 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".

Equivalent conditions

Many conditions have been shown to be equivalent to being sequential. Here are a few:

• X is sequential.
• X is the quotient of a first countable space.
• X is the quotient of a metric space.
• 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 (x_n) in X converging to x, we have (f(x_n)) converging to f(x).

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.

Categorical properties

The category of all sequential spaces is closed under the following operations:

• Quotients
• Continuous closed or open images
• Sums
• Inductive limits
• Open and closed subspaces

The category of all sequential spaces is not closed under the following operations:

• Continuous images
• Subspaces
• Products

Since they are closed under topological sums and quotiens, 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 quotiens and containing the metrizable spaces).

See also

• axioms of countability
• first-countable space

References

• Engelking, R., General Topology, PWN, Warsaw, (1977).
• Franklin, S. P., "Spaces in Which Sequences Suffice", Fund. Math. 57 (1965), 107-115.
• Franklin, S. P., "Spaces in Which Sequences Suffice II", Fund. Math. 61 (1967), 51-56.
• Goreham, Anthony, "Sequential Convergence in Topological Spaces
• A.V. Arkhangel'skii and L.S. Pontryagin, General Topology I, Springer-Verlag, New York (1990) ISBN 3-540-18178-4.