nLab sigma-pretopological space

Idea

A pretopological space is a convergence space which has all intersections of convergent filters to an element also converge to that element. Thus, a $\sigma$-pretopological space should be a convergence space which should have countable intersections of convergent filters to an element converge to that element.

Definition

A $\sigma$-pretopological space $S$ is a set $S$ with a relation $F \to x$ from the set of filters on $S$ to $S$ itself, satisfying the following properties

• Centred: The free ultrafilter at $x$ (the collection of all sets that $x$ belongs to) converges to $x$:
  $$\{ A \;|\; x \in A \} \to x .$$
• Isotone: If $F$ converges to $x$ and $G$ refines $F$, then $G$ converges to $x$:
  $$F \to x \;\Rightarrow\; F \subseteq G \;\Rightarrow\; G \to x .$$
• Countably directed: The intersection of a countable family of filters that converge to $x$ itself converges to $x$:
  $$\forall n \in \mathbb{N}.(F_n \to x) \; \Rightarrow \; \left(\bigcap_{n \in \mathbb{N}} F_n\right) \to x .$$

 Related concepts

• convergence space

• pretopological space