Sequential space

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.^[1]