$X$ 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?

Suppose $X$ isn't connected. Then there are closed sets $V, W\subsetneq X$ with $V\cup W=X, V\cap W=\emptyset$. Then we have relatively closed sets $V\cap S\subseteq S, W\cap S\subseteq S$ with $(W\cap S)\cap (V\cap S)=\emptyset$ and we also have $W\cap S\subsetneq S$ and $V\cap S\subsetneq S$, then if e.g $W\cap S= S$ was correct, we had $\bar{S}\subseteq W\subsetneq X$.
So all in all we see $S$ isn't connected, a contradiction.
