$X = (X,T)$ is topological space and $A \subset X$. I want to prove, that $\bar A = \cap C_i$ where $C_i$ - are closed sets, which contains A.
$(\Rightarrow)$ We will show, that $\bar A \subseteq \cap C_i$. Let $x \in \bar A$. Let's consider two cases:
a) If $x \in \bar A$ and $C_i$ closed sets, that contains $A \Rightarrow \cap C_i$ will contain $A \Rightarrow\cap C_i$ will contain $x$ (by definition of intersection)
b)If $x$ is a limit point of $A$: By definition closed set contains all it's limit points. If $C$ contains set $A$ and $C$ contains a l.p of $C\Rightarrow$ it contains a l.p of $A$. If every set $C$ contains limit point of $A$ - then intersection contains l.p. of $A$ as well.
Hence, $\bar A \subseteq \cap C_i$
$(\Leftarrow)$ we will show, that $A \supseteq \cap C_i$
Suppose $x \in \cap C_i$:
a) Because every $C_i$ contains $A$, then $\cap C_i$ contains $A\Rightarrow x \in A$.
I have problem with the reverse direction. Give me some hints please
It's of course more the idea how the proof should like. If the idea is correct, I would like to ask you how should i formalize my arguments.
If it's not correct, what shoud I correct?