Show that if $C_1$ and $C_2$ are two distinct components of a space X, then $C_1\cap C_2$ = $\emptyset$

Question

This is part of my proof:Suppose $C_1 \cap C_2\neq \emptyset$,then there exists some element $x$ in $C_1 \cap C_2$.Since $C_1$ and $C_2$ are connected and closed,then......

Can someone continue the proof? I think the last step would be showing some contradiction

Answer 1

$C_1 \cup C_2$ is also connected, contradicting the maximality of components (a component is a maximimally connected subset; any larger subset is no longer connected).