I am new to topology and I am stuck at this basic question. Let $Y$ be a subspace of $X$ and A be a subset of Y. I need to show that
A is connected in Y $\iff$ A is connected in X.
$\Rightarrow$ is easy. If $A = C \cup D$, where $C$ and $D$ are disjoint open sets in X, then $Y \cap C = C$ and $Y \cap D = D$ and we get $C$ and $D$ are open sets in Y.
Can you help me with $\Leftarrow$?