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.
 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 of X is sequentially closed if, whenever (x_{n}) is a sequence in F converging to x, then x must also be in F.
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. The sequentially open sets form a topology, which contains the original topology of the space and has the same notion of sequential convergence.
 Let U be sequentially open. Let us show that its complement F= X\U is sequentially closed, i.e. that a convergent sequence (x_{n})_{n∈ℕ} of elements of F has its limit in F.
 By absurd, if , then . This contradicts the fact that all x_{n} are supposed to be in F.
 Conversely if F is sequentially closed, let us show that its complement U= X\F is sequentially open.
 Let (x_{n})_{n∈ℕ} be a sequence in X such that and suppose by absurd that for any , i.e. . Define by induction the subsequence (x_{φ(n)})_{n∈ℕ} of elements of F: set φ(0)= k_{0} and then φ(n+1)= k_{φ(n)+1} , i.e. x_{φ(0)}:= x_{k0} and x_{φ(n+1)} = x_{kφ(n)+1}. It is convergent as a subsequence of a convergent sequence, and all its elements are in F. Hence the limit has to be in F, which contradicts x∈U. The sequence is therefore eventually in U.
 Let us show that the set of sequentially open subsets is a topology, i.e. ∅ and X are sequentially open, arbitrary unions of sequentially open subset is sequentially open and finite intersections of sequentially open subsets is sequentially open.
 Any empty sequence satisfies any property and any sequence in X is eventually in X. Let (U_{i})_{i∈I} be a family of sequentially open subsets, and (x_{n})_{n∈ℕ} a sequence in X converging to x∈U. x element of the union means and by sequential openness, the sequence is eventually in U_{i0}. Finally, if is a finite intersection of sequentially open subsets, then a sequence converging to x∈V eventually converges to each of the U_{i}, i.e. . Taking , one has .
 The generated sequential topology is finer than the original one, i.e. if U is open, then it is sequentially open.
 Let (x_{n})_{n∈ℕ} be a sequence in X converging to x∈U. Since U is open, it is a neighborhood of x and by definition of convergence, there exists .
A sequential space is a space X satisfying one of the following equivalent conditions:
 Every sequentially open subset of X is open.
 Every sequentially closed subset of X is closed.
 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 .[1]
That is, the original topology can be reconstructed by knowing the convergent sequences.
 (): Assume that any sequentially open subsets is open and let F be sequentially closed. It is proved above that the complement U=X\F is sequentially open and thus open so that F is closed. The converse is similar.
 (): Contraposition of 2. says that «S not closed implies S not sequentially closed», hence there exists a sequence of elements of S that converges to a point outside of S. Since the limit is necessarily adherent to S, it is in the closure of S.
 Conversely, let us suppose by absurd that a subset S:=F is sequentially closed but not closed. By 3., there exists a sequence in F that converges to a point in , i.e. the limit lies outside F. This contradicts sequential closedness of F.
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 , to be , and for a limit ordinal , to be . Then there is a smallest ordinal such that , and for this , is called the transfinite sequential closure of . (In fact, we always have , where is the first uncountable ordinal.) Taking the transfinite sequential closure solves the idempotency problem above.
The smallest such that for each is called sequential order of the space X.[2] This ordinal invariant is welldefined for sequential spaces.
Fréchet–Urysohn space
Topological spaces for which sequential closure is the same as ordinary closure are known as Fréchet–Urysohn spaces (such a space is also said to be Fréchet). 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 firstcountable 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.[3][4]
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 (aka Stan 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 firstcountable 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 firstcountable space is sequential, hence each secondcountable space, metric space, and discrete space is sequential. Further examples are furnished by applying the categorical properties listed below. For example, every CWcomplex is sequential, as it can be considered as a quotient of a metric space.
There are sequential spaces that 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 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".
Equivalent conditions
Many conditions have been shown to be equivalent to X being sequential. Here are a few:
 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 full subcategory Seq of all sequential spaces is closed under the following operations in Top:
 Quotients
 Continuous closed or open images
 Sums
 Inductive limits
 Open and closed subspaces
The category Seq is not closed under the following operations in Top:
 Continuous images
 Subspaces
 Finite products
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 (i.e., the smallest class of topological spaces closed under sums and quotients and containing the metrizable 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. 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, CWcomplexes, and differentiable manifolds and that is closed under colimits, quotients, and other "certain reasonable identities" that Norman Steenrod described as "convenient".
References
 Arkhangel'skii, A.V. and Pontryagin L.S., General Topology I, definition 9 p.12

 Arhangel'skiĭ, A. V.; Franklin, S. P. (1968). "Ordinal invariants for topological spaces". Michigan Math. J. 15 (3): 313–320. doi:10.1307/mmj/1029000034.
 Engelking 1989, Example 1.6.18
 Ma, Dan. "A note about the Arens' space". Retrieved 1 August 2013.
 Arkhangel'skii, A.V. and Pontryagin, L.S., General Topology I, SpringerVerlag, New York (1990) ISBN 3540181784.
 Booth, P.I. and Tillotson, A., Monoidal closed, cartesian closed and convenient categories of topological spaces Pacific J. Math., 88 (1980) pp. 35–53.
 Engelking, R., General Topology, Heldermann, Berlin (1989). Revised and completed edition.
 Franklin, S. P., "Spaces in Which Sequences Suffice", Fund. Math. 57 (1965), 107115.
 Franklin, S. P., "Spaces in Which Sequences Suffice II", Fund. Math. 61 (1967), 5156.
 Goreham, Anthony, "Sequential Convergence in Topological Spaces"
 Steenrod, N.E., A convenient category of topological spaces, Michigan Math. J., 14 (1967), 133152.