Definition of accumulation point of a net

Question

Let $(x_i)_{i \in I}$ be a net in a topological space $X$. An accumulation point of the net $(x_i)_{i \in I}$ is an element of the intersection $ \bigcap_{F \in \mathcal{F}} \overline{F}$ where $$ \mathcal{F} =\big\{F \subset X:\text{ there exists } i_0 \in I \text{ such that }\{x_{i} : i \geq i_0\} \subset F \big\}. $$ Is it true ? I does not want a proof. A reference is welcome if there exists.

Answer 1

Yes, this is true. The proof is not difficult, so I will write it:

Assume $y$ is a limit point. Take any $F \in \mathcal{F}$ and assume that $x_i \in F$ for all $i \ge i_0$. Let $U$ be a neighborhood of $y$. Then there must be $j \ge i_0$ such that $x_j \in U$, because $y$ is a limit point. Hence $x_j \in U \cap F$. In particular, $y \in \overline{F}$, because every neighborhood of $y$ intersects $F$.

Conversely, assume that $y$ is not a limit point. Then there is a neighborhood $U$ of $y$ and some $i_0$ such that if $i \ge i_0$, then $x_i \notin U$. Define $F:= \{x_i \mid i \ge i_0\}$. Then $U \cap F$ is empty, hence $y \notin \overline{F}$.