I have the following problem:
Let $\mathfrak{F}$ be a filter with associated net $(p_i)_{i\in I_\mathfrak{F}}$. Show that $p\in M$ is a cluster point of $\mathfrak{F}$ iff $p$ is a cluster point of the net $(p_i)_{i\in \mathfrak{F}}$.
Definiton of the associated net If $\mathfrak{F}$ is a filter on M then $I_\mathfrak{F}=\{(A,p):A\in\mathfrak{F}, p\in A\}$ is directed by $(A,p)\leq (B,q)$ if $B\subset A$. Then we have a net $$I_\mathfrak{F}\rightarrow M;\,\,\,(A,p)\mapsto p$$ this is called the associated the of the filter $\mathfrak{F}$
My Idea for one direction was the following but I somehow had problems with this indices and maybe also in understanding the whole thing here.
$\Rightarrow$ Let $p\in M$ be a cluster point of $\mathfrak{F}$. This means by definition that for all $U\in \mathfrak{U}(p)$ neighbourhoods of $p$ and for all $A\in \mathfrak{F}$ $$A\cap U\neq \emptyset$$Now let us take $U\in \mathfrak{U}(p)$ and take $i=(B,p_i)\in I_\mathfrak{F}=\{(A,p):A\in \mathfrak{F}, p\in A\}$, then $i\mapsto p_i\in B$. But then $p_i\in U$ since p is a cluster point of $\mathfrak{F}$. Now take $j=(A,p_j)\in I_\mathfrak{F}$ where $p_j\geq p_i$, then $p_j\in U$ and thus it is a cluster point of the net.
Could someone take a look and help me maybe?
Thank you
You're onto the right idea but need to be more careful about what exactly you need to prove.
To prove $p$ is a cluster point of the net $I_\mathfrak{F}$, as you've correctly seen you need to show that for each neighborhood $U$ of $p$ and $i_0\in I_\mathfrak{F}$, there exists some $i\geqslant i_0$ such that $p_i\in U$.
The mistake you make above is that (writing $i_0=(A,p_{i_0})$) you don't necessarily know that $p_{i_0}\in U$. But what you know is there is some $A\in\mathfrak{F}$ such that $A\cap U\neq\emptyset$. Thus if you take $x\in A\cap U$, then you can take $i=(A,x)$.