properties of connected set

Question

If $C $ is a connected set in a metric space $X$ & $C$ intersects both $A$ and $X\cap A^c (A\subseteq X)$ then can it be concluded that $C\cap \delta A\neq\phi$ ?

Answer 1

Suppose that $C\cap\delta A=\emptyset$. Then $C\cap \overline {A^c}\cap\bar A=\emptyset$

For each $x\in A\cap C$ we can say that $x\notin \overline{A^c}$, that is, we can take $U_x$, an open neighbourhood of $x$ that doesn't intersect $A^c$.

Let $$U=\bigcup_{x\in A\cap C}U_x$$

Make the same reasoning to get $$V=\bigcup_{x\in A^c\cap C}V_x$$

$U,V$ are open, nonpempty disjoint sets that cover $C$, a contradiction with its conectedness.

Note that $X$ needn't be a metric space, just topological.