Show that the filter $\mathscr F$ has $x$ as a cluster point iff $x \in\bigcap_{F \in \mathscr F } \overline F$.
For the the Proof of the 1st direction $(\Rightarrow)$ : Let the filter $\mathscr F$ has $x$ as a cluster point so every element $F$ of $\mathscr F$ intersect every $U \in $ $\mathscr U_x$ where $\mathscr U_x$ is the nbd system ,since every $U$ intersects $F$ , so $x \in\overline F \Rightarrow x \in \cap \overline F$.
Now for the other direction $(\Leftarrow)$ : Let $x \in \cap \overline F$ so this mean that $x$ belongs to each $\overline F$ then every nbd $U$ of $x$ intersect $F$, and $F$ is elements of $\mathscr F$, then $x$ is a cluster point of $\mathscr F$.
If any one can tell me that my proof is correct or not ?