if a convergent sequence is contained in a closed set
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.)
Sequential spaces are those topological spaces for which nets of countable length (i.e., sequences) suffice to describe the topology.
[1] 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.
which defines a map, the sequential closure operator, on the power set of
One can obtain an idempotent sequential closure via transfinite iteration: for a successor ordinal
is sequential if it satisfies any of the following equivalent conditions: By taking
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.
There exist topological vector spaces that are sequential but not
is called Fréchet–Urysohn if it satisfies any of the following equivalent conditions: 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.
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.
As a quotient of a metric space, the result is sequential, but it is not first countable.
A Hausdorff topological vector space is sequential if and only if there exists no strictly finer topology with the same convergent sequences.
of smooth functions, as discussed in the article on distributions, are both widely-used sequential spaces.
[10][11] More generally, every infinite-dimensional Montel DF-space is sequential but not Fréchet–Urysohn.
[12][13] 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.
are Montel spaces[15] 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).
[10][16] Every sequential space has countable tightness and is compactly generated.
is a continuous open surjection between two Hausdorff sequential spaces then the set
is a surjective map (not necessarily continuous) onto a Hausdorff sequential space
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.
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.
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".