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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 ?

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    @AndreasBlass thank you i will delete them from the proof. ;@leo Let $\mathbf E \subset \mathbf X$ , a point $x \in \mathbf X$ is a cluster point at $\mathbf E$ iff every nbd of$x$contains a point of $\mathbf E$ different from x.2012-11-18

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Write it down more calmly, using more sentences etc.:

Suppose $x$ is a cluster point of $\mathscr F$. We want to show that $x \in\bigcap_{F \in \mathscr F } \overline F$, and so pick an arbitrary $F \in \mathscr F$. To see $x \in \overline F$, we pick any open neighbourhood $O$ of $x$, and we need to see that $O$ intersects $F$. But this is clear from the definition of $x$ being a cluster point got $\mathscr F$. The reverse is similar.

If you want to see it more as a logical fact plus definitions:

$x$ is a cluster point of $\mathscr F$ means by definition $\forall O \in \mathscr{U}_{x} : \forall F \in \mathscr{F}: O \cap F \neq \emptyset$ which is the same as $\forall F \in \mathscr{F}: ( \forall O \in \mathscr{U}_{x} : O \cap F \neq \emptyset)$, and the statement in brackets is by (a) definition the meaning of $x \in \overline{F}$, so it also says $ \forall F \in \mathscr{F}: x \in \overline{F} $, which by definition of intersection is just $x \in \bigcap_{F \in \mathscr F} \overline{F} $.

So the fact is just a simple restatement of the definions of closure, intersection and cluster point.