a question about the definition of the Chabauty topology for discrete groups

Question

Let $G$ be a discrete group and let $S(G)$ the set of subgroups equipped with the following topology:

A net $(H_i)_i\subset G$ of subgroups converges in the Chabauty topology to a subgroup $H$ of $G$ if

(1) every $h\in H$ eventually belongs to $H_i$

(2) for every subnet $(H_{j_k})_k$ of $(H_i)_i$ it is $\bigcap_k H_{j_k}\subset H$.

In (1), what does "eventually belongs" in (1) mean (i.e. what is meant with (1))? I am a non-native speaker in english.

Answer 1

The net $I$ is a directed set with order relation $\geq$. The statement in (1) means: $$ \forall h \in H \exists i_0 \in I \forall i \in I: (i \geq i_0 \Rightarrow h \in H_i). $$