In topology and related fields of mathematics, a sequential space is a topological space whose topology can be completely characterized by its convergent/divergent sequences. They can be thought of as spaces that satisfy a very weak axiom of countability, and all first-countable spaces (notably metric spaces) are sequential.

In any topological space ${\displaystyle (X,\tau ),}$ if a convergent sequence is contained in a closed set ${\displaystyle C,}$ then the limit of that sequence must be contained in ${\displaystyle C}$ as well. Sets with this property are known as sequentially closed. 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.) Said differently, any topology can be described in terms of nets (also known as Moore–Smith sequences), but those sequences may be "too long" (indexed by too large an ordinal) to compress into a sequence. Sequential spaces are those topological spaces for which nets of countable length (i.e., sequences) suffice to describe the topology.

Any topology can be refined (that is, made finer) to a sequential topology, called the sequential coreflection of ${\displaystyle X.}$

The related concepts of Fréchet–Urysohn spaces, T-sequential spaces, and ${\displaystyle N}$-sequential spaces are also defined in terms of how a space's topology interacts with sequences, but have subtly different properties.

Sequential spaces and ${\displaystyle N}$-sequential spaces were introduced by S. P. Franklin.

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.

Let ${\displaystyle X}$ be a set and let ${\displaystyle x_{\bullet }=\left(x_{i}\right)_{i=1}^{\infty }}$ be a sequence in ${\displaystyle X}$; that is, a family of elements of ${\displaystyle X}$, indexed by the natural numbers. In this article, ${\displaystyle x_{\bullet }\subseteq S}$ means that each element in the sequence ${\displaystyle x_{\bullet }}$ is an element of ${\displaystyle S,}$ and, if ${\displaystyle f:X\to Y}$ is a map, then ${\displaystyle f\left(x_{\bullet }\right)=\left(f\left(x_{i}\right)\right)_{i=1}^{\infty }.}$ For any index ${\displaystyle i,}$ the tail of ${\displaystyle x_{\bullet }}$ starting at ${\displaystyle i}$ is the sequence $${\displaystyle x_{\geq i}=(x_{i},x_{i+1},x_{i+2},\ldots ){\text{.}}}$$ A sequence ${\displaystyle x_{\bullet }}$ is eventually in ${\displaystyle S}$ if some tail of ${\displaystyle x_{\bullet }}$ satisfies ${\displaystyle x_{\geq i}\subseteq S.}$

Let ${\displaystyle \tau }$ be a topology on ${\displaystyle X}$ and ${\displaystyle x_{\bullet }}$ a sequence therein. The sequence ${\displaystyle x_{\bullet }}$ converges to a point ${\displaystyle x\in X,}$ written ${\displaystyle x_{\bullet }{\overset {\tau }{\to }}x}$ (when context allows, ${\displaystyle x_{\bullet }\to x}$), if, for every neighborhood ${\displaystyle U\in \tau }$ of ${\displaystyle x,}$ eventually ${\displaystyle x_{\bullet }}$ is in ${\displaystyle U.}$ ${\displaystyle x}$ is then called a limit point of ${\displaystyle x_{\bullet }.}$