Question

Q5 Let F be closed set. Prove that ∀ A subset and equel to X；line over (F∩A)subset and equel to (F∩(lin overA)).

Answer 1

Let $F$ be closed set. Let us prove that $\forall A\subseteq X$ we have that $\overline{F\cap A}\subseteq F\cap \overline{A}$.

Let $x\in\overline{F\cap A}$ . Then $x$ is a cluster point of the set $F\cap A$. This means that each neighbourhood of $x$

contains some element $y\in F\cap A$. Therefore, $y\in F$ and $y\in A$, and each neighbourhood of $x$ contains an element $y\in F$ and each neighbourhood of $x$ contains an element $y\in A$

Consequently, $x\in \overline{F}$ and $x\in \overline{A}$. Since $F$ is a closed set, $\overline{F}=F$. We conclude that $x\in F$, and thus $x\in F\cap\overline{A}$.