is identical to the sequential closure of
The property is named after Maurice Fréchet and Pavel Urysohn.
if there exists a positive integer
is called sequentially open if every sequence
is called sequentially closed if
The complement of a sequentially open set is a sequentially closed set, and vice versa.
denote the set of all sequentially open subsets of
sequentially closed), which implies that
is a strong Fréchet–Urysohn space if for every point
The above properties can be expressed as selection principles.
The spaces for which the converses are true are called sequential spaces; that is, a sequential space is a topological space in which every sequentially open subset is necessarily open, or equivalently, it is a space in which every sequentially closed subset is necessarily closed.
can be used as a "test" to determine whether or not any given subset is open (or equivalently, closed) in
; or said differently, sequential spaces are those spaces whose topologies can be completely characterized in terms of sequence convergence.
In any space that is not sequential, there exists a subset for which this "test" gives a "false positive.
is a topological space then the following are equivalent: The characterization below shows that from among Hausdorff sequential spaces, Fréchet–Urysohn spaces are exactly those for which a "cofinal convergent diagonal sequence" can always be found, similar to the diagonal principal that is used to characterize topologies in terms of convergent nets.
In the following characterization, all convergence is assumed to take place in
then there exist strictly increasing maps
is infinite) because if it is finite then Hausdorffness implies that it is necessarily eventually constant with value
with the desired properties is readily verified for this special case (even if
[3][4] If a Hausdorff locally convex topological vector space
A metrizable locally convex topological vector space (TVS)
[6] Direct limit of finite-dimensional Euclidean spaces The space of finite real sequences
where the latter is a subset of the space of sequences of real numbers
(with it usual Euclidean topology).
(in the Euclidean norm) centered at the origin.
Montel DF-spaces Every infinite-dimensional Montel DF-space is a sequential space but not a Fréchet–Urysohn space.
denote the Schwartz space and let
denote the space of smooth functions on an open subset
where both of these spaces have their usual Fréchet space topologies, as defined in the article about distributions.
The strong dual spaces of both