nLab sequential preconvergence space

Idea

By a sequential preconvergence space we shall mean the most general kind of topological space where the notion of convergence and limit for sequences makes sense. The possibly more evident term “sequential convergence space” for this notion is already taken to mean a synonym of subsequential space, but that definition is not the most general definition for which the notions of convergence and limit makes sense for sequences.

 Definition

A set $S$ is a sequential preconvergence space if it comes with a binary relation $x \to c$ between the set of all sequences in $S$ and $S$ itself, for $x \in S^\mathbb{N}$ and $c \in S$. A sequential preconvergence space is sequentially Hausdorff if the binary relation is a functional relation, and every sequentially Hausdorff sequential preconvergence space has a partial function

$$\lim_{n \to \infty} (-)_n:(\mathbb{N} \to S) \to S$$

 See also

• limit of a sequence
• sequentially Hausdorff space
• subsequential space
• preconvergence space
• metric space
• uniform space
• Cauchy space
• Booij premetric space