Connected components Let $X$ be a topological space. We define $x\sim y$ if and only if $C_x=C_y$ (their connected components).
Let $X/\sim$ be the quotient space. 
I want to prove that $C_{\bar{x}}=\{\bar{x}\}$ for each $\bar{x}\in X/\sim$.
By contradiction, I suppose that $\{\bar{x},\bar{y}\}\subseteq C_{\bar{x}}$ for some $\bar{y}\neq\bar{x}$. This is $C_x\neq C_y$ in $X$. 
I don't know how to continue from here.
Any hint? Thanks.

If $\overline{y}\in C_\overline{x}$, then $x\sim y$ so you get $\overline{x}=\overline{y}$.
