nLab quasi-net

Definition

In non-archimedean analytic geometry, one considers:

Given a topological space, a quasinet in $X$ is a set $\tau = \{V_i\}$ of subsets of $X$ such that for each point $x\in X$ there exists a finite number of such subsets $V_{i_1}, \cdots, V_{i_n} \in \tau$ such that

1. $x\in V_{i_1} \cap V_{i_2} \cap \cdots \cap V_{i_n}$ (the point is in their intersection);

2. $V_{i_1} \cup \cdots \cup V_{i_n}$ is a neighbourhood of $x$.

(e.g. Berkovich 09, def. 3.1.1.)

A quasinet $\tau$ is called a net if for all $U,V \in \tau$ then the restriction $\tau|_{U \cap V}$ is a quasinet of $U \cap V$.

Examples

• A Berkovich analytic space is a locally Hausdorff topological space equipped with an atlas by affinoid domains such that the underlying analytic spectrum topological spaces of these form a net of compact subsets on $X$.

References

• Vladimir Berkovich, Non-archimedean analytic spaces, lectures at the Advanced School on $p$-adic Analysis and Applications, ICTP, Trieste, 31 August - 11 September 2009 (pdf)