Sequentially continuous function

A sequentially continuous function is a function between topological spaces that preserves the limits of convergent sequences. For topological spaces (X) and (Y), a function

[ f\colon X\longrightarrow Y ]

is sequentially continuous at (x\in X) when every sequence ((x_n){n\in\mathbb N}) converging to (x) has an image sequence ((f(x_n)){n\in\mathbb N}) converging to (f(x)). The function is sequentially continuous when this condition holds at every point of its domain.

Every continuous function is sequentially continuous. The converse holds for metric spaces, first-countable spaces, and, more generally, sequential spaces. It fails for arbitrary topological spaces because sequences do not always detect the entire topology.

Definition

A sequence ((x_n)) in a topological space (X) converges to (x\in X) if, for every neighborhood (U) of (x), there exists (N\in\mathbb N) such that

[ n\geq N\quad\Longrightarrow\quad x_n\in U. ]

Sequential continuity of (f\colon X\to Y) at (x) is therefore expressed by the implication

[ x_n\longrightarrow x \quad\Longrightarrow\quad f(x_n)\longrightarrow f(x). ]

This definition depends only on convergent sequences and their limits. It does not directly refer to inverse images of open sets, which form the usual definition of topological continuity.

Continuity implies sequential continuity because, whenever (V) is a neighborhood of (f(x)), continuity supplies a neighborhood (U) of (x) satisfying (f(U)\subseteq V). A sequence converging to (x) is eventually contained in (U), so its image is eventually contained in (V).

Characterization by sequentially open sets

A subset (A\subseteq X) is sequentially open when every sequence converging to a point of (A) is eventually contained in (A). Equivalently, (A) is sequentially open if

[ x_n\longrightarrow x\in A \quad\Longrightarrow\quad x_n\in A\text{ for all sufficiently large }n. ]

A function (f\colon X\to Y) is sequentially continuous exactly when (f^{-1}(V)) is sequentially open in (X) for every open subset (V\subseteq Y). Indeed, preservation of sequential limits implies eventual membership in each neighborhood of the image limit. Conversely, the inverse-image condition forces the image of every convergent sequence to be eventually contained in each neighborhood of the appropriate image point.

Every open set is sequentially open, but a sequentially open set need not be open. This difference accounts for the possible separation between continuity and sequential continuity.

The sequentially open subsets of (X) form a topology finer than the original topology. The resulting space is called the sequentialization of (X), commonly denoted (sX). A map (f\colon X\to Y) is sequentially continuous precisely when the corresponding map

[ f\colon sX\longrightarrow Y ]

is continuous.

Equivalence in first-countable spaces

Suppose that (X) is first countable and that (f\colon X\to Y) is not continuous at (x). There is then a neighborhood (V) of (f(x)) such that every neighborhood (U) of (x) contains a point whose image does not belong to (V).

Let

[ U_1\supseteq U_2\supseteq U_3\supseteq\cdots ]

be a decreasing countable neighborhood base at (x). For each (n), a point (x_n\in U_n) can be selected with (f(x_n)\notin V). The sequence ((x_n)) converges to (x), while ((f(x_n))) cannot converge to (f(x)), since it never enters the neighborhood (V). Consequently, sequential continuity implies continuity whenever the domain is first countable.

This argument applies in particular to metric spaces, because the balls centered at (x) with radii (1/n) form a countable neighborhood base. No countability condition on the codomain is required.

Failure of the converse in general

Let (\omega_1) be the first uncountable ordinal, and equip the ordinal interval

[ [0,\omega_1] ]

with its order topology. Define a function into the discrete two-point space ({0,1}) by

[ f(\alpha)= \begin{cases} 0,&\alpha<\omega_1,\ 1,&\alpha=\omega_1. \end{cases} ]

The function is not continuous at (\omega_1). The singleton ({1}) is open in the codomain, but its inverse image ({\omega_1}) is not open in the ordinal interval.

Nevertheless, (f) is sequentially continuous. Any countable sequence of ordinals below (\omega_1) has a countable supremum, which remains strictly below (\omega_1). Such a sequence therefore cannot converge to (\omega_1). More generally, every sequence converging to (\omega_1) is eventually equal to (\omega_1), and its image is consequently eventually equal to (1).

This example isolates the underlying obstruction: the neighborhoods of (\omega_1) contain information that no countable sequence of smaller ordinals can recover. Nets and filters do detect this information, which is why preservation of all convergent nets is equivalent to continuity without additional assumptions on the spaces.

Sequential spaces

A topological space (X) is sequential when every sequentially open subset of (X) is open. Equivalently, a subset of (X) is closed whenever it contains the limits of all convergent sequences whose terms lie in that subset.

For a sequential domain (X), every sequentially continuous map (f\colon X\to Y) is continuous. The characterization by inverse images gives this result directly: inverse images of open subsets of (Y) are sequentially open in (X), and sequentiality makes them open.

Sequential spaces include all first-countable spaces, but the class is larger. In particular, quotients of first-countable spaces can remain sequential even when first countability is lost. S. P. Franklin established the quotient characterization under which a space is sequential exactly when it is a quotient of a metric space.

A Fréchet–Urysohn space satisfies a stronger condition. Whenever (x) belongs to the closure of a subset (A), some sequence of points of (A) converges to (x). Every Fréchet–Urysohn space is sequential, although sequential spaces need not have this pointwise closure property.

Structural properties

Sequentially continuous maps are closed under composition. If (f\colon X\to Y) and (g\colon Y\to Z) preserve convergent sequences, then

[ x_n\to x \quad\Longrightarrow\quad f(x_n)\to f(x) \quad\Longrightarrow\quad g(f(x_n))\to g(f(x)). ]

The identity map on any topological space is sequentially continuous, so topological spaces and sequentially continuous maps form a category. This category has more morphisms than the ordinary category of topological spaces and continuous maps whenever nonsequential domains are present.

Restrictions of sequentially continuous functions remain sequentially continuous. For a function into a product space, sequential continuity is equivalent to sequential continuity of every coordinate function, since convergence in the product topology is characterized coordinatewise.

When the codomain is a topological group, sequentially continuous functions are also compatible with the group operations. Pointwise products and inverses preserve sequential continuity because multiplication and inversion are continuous and hence preserve convergent sequences.

Historical development

The systematic use of sequences in topology developed from Maurice Fréchet's work on metric spaces and abstract convergence during the early twentieth century. In metric settings, the equivalence between the neighborhood definition of continuity and preservation of sequential limits made a separate term unnecessary.

The distinction became substantive as general topology moved beyond countability assumptions. In 1964, You Watanabe analyzed preservation of convergent sequences as an independent property of maps on non-first-countable spaces and used ordinal domains to separate it from open-set continuity. The formulation placed sequential continuity within the broader study of topologies determined only partially by countable convergence.

Subsequent work on sequential spaces supplied an intrinsic description of the domains on which the two notions coincide. The subject is now treated as part of the relationship among convergence structures, countability axioms, and categorical modifications of topological spaces.