Let $X $ be a topological space and $Y $ a Hausdorff subspace of $X $ such that for every connected component $C $ of $X $ the set $C\cap Y $ is finite. How can we show that $Y $ is totally disconnected?
My proof: Let $A $ be a connected component of $Y $. Then there is a connected component $C $ of $X $ such that $A\subseteq C $. This show that $A $ is finite and since $Y $ is Hausdorff, $A $ must be a single point. Hence $Y $ is totally disconnected. IS THIS PROOF TRUE ?