$X$ topological space and $S\subseteq X$ and $\overline{S}=X$. Show that if $S$ is connected, then $X$ is also $X$ topological space and $S\subseteq X$ and $\overline{S}=X$. Show that if $S$ is connected, then $X$ is also.
First of all, what is the intuition behind this? For example, what is a subset $S$ such that $\overline{S} = X$?. I can't even think of such thing... Is it a space that is almost $X$? And then when we take the closure and it's $X$. Well, what argument should be used to argue that $X$ will also be connected?

Assume that there is a continuous surjection $\varphi:X\rightarrow\{0,1\}$ and we let without loss of generality that some $s\in S$ be such that $\varphi(s)=0$. As $S$ is connected, we have $\varphi[S]=\{0\}$. Then for any $x\in X-S$, we could find some net $\{x_{\delta}\}\subseteq S$ such that $\varphi(x)=\lim_{\delta}\varphi(x_{\delta})=\lim_{\delta}0=0$, it violates the condition that some $x\in X$ is such that $\varphi(x)=1$.
