Let $(Y, \tau)$ be a locally connected topological space and $A\subset Y$. Let $C$ be a component of $A$. Show that $\partial C\subset \partial A$.
I proved that $Int_{Y}(C)= C\cap Int_{Y}(A)$. Using that, $x\in \partial C =\overline{C}\setminus Int_{Y}(C)= \overline{C}\setminus (C\cap Int_{Y}(A))$, but $$\overline{C}\setminus (C\cap Int_{Y}(A))= \overline{C}\cap (C\cap Int_{Y}(A))^{c}= (\overline{C}\cap C^{c})\cup (\overline{C}\cap Int_{Y}(A)^{c})$$ I want to show that $x\notin (\overline{C}\cap C^{c})$. If $x\in (\overline{C}\cap C^{c})$, then $x$ is a limit point, Using this can I arrive at a contradiction?. or there is direct proof?. thanks