Suppose that $C$ is a connected subset of $X$, and $U$ an open set in $X$. Also, we have that $C$ contains a point in $X$ and a point not in $X$. Then how do we show that the following is true: $C \cap \partial U \neq \varnothing?$
My second question is that what is a good example in $\mathbb{R}^2$ of a connected set $C$ with an open set (disk) $U$ such that $C$ contains a point inside $U$ and a point in the complement of cl$(U)$, i.e. the closure of $U$, and some component of $U \cap C$ misses $\partial U$?
For the first question, right now the only thing that comes to mind is a proposition that states for $A \subset X$, if $C$ is a connected subspace of $X$ such that $C \cap A \neq \varnothing, C \cap (X-A) \neq \varnothing,$ then $C \cap \partial A \neq \varnothing$. In this case, however, we don't know if $A$ is open, and nowhere is it mentioned that $C$ contains a point in $X$ and a point not in $X$. So how should we proceed? I would also really appreciate some guidance on the second question for a suitable example and why it works.