let $(X,d)$ be metric space and y $A$,subsets of $X$

Question

I want to do this topology problem:

let $(X,d)$ be metric space and y $A$ subsets of
$X$ such that $E\cap A \neq \varnothing$ and $E\cap (X\smallsetminus A) \neq\varnothing$. if $E$ is connected, prove $E\cap \partial A\neq \varnothing$.

I have tried to reason by reduction to the absurd and applying definition but I don't know if that path is viable. Could you give me a suggestion? Please

Answer 1

You probably mean that $E$ is connected. Plus there is no $B$.

The essence is that $E \cap \partial A = \emptyset$, then $(X\setminus A) \cap E$ and $E \cap A$ form a non-trivial partition of $E$. Try to show that.