Connected $T_3$ space I wanted to prove that a connected $T_3-$space containing atleast two points must be uncountable.
My attempt is as follows:
Let $X$ be a connected $T_3-$ space and let $x,y\in X$. Then there exist disjoint open sets $G,H$ containing $x$ and $y$ respectively. Since $X$ is connected, therefore, $X\neq G\cup H$. Thus $\exists z\in X$ such that $z\notin G\cup H$. 
I couldn't proceed further. Please help.

The only proofs that I can see at the moment are indirect: suppose that $X$ is countable and $T_3$, and show that $X$ cannot be connected. A countable space is clearly Lindelöf, and it’s a standard basic result that a Lindelöf $T_3$-space is $T_4$, so $X$ is $T_4$ and therefore Tikhonov. Now let $x$ and $y$ be distinct points of $X$; since $X$ is Tikhonov, there is a continuous function $f:X\to[0,1]$ such that $f(x)=0$ and $f(1)=1$. Now use the fact that $f[X]$ is countable to find disjoint non-empty open sets $U$ and $V$ in $X$ such that $U\cup V=X$.
