Two questions on which I am stuck:
1.Show that a connected normal space having more than one point is uncountable.
2.Show that a connected regular space having more than one point is uncountable.
Let $X$ be a connected normal space.Then we can always find two disjoint closed sets in $X$ say $\{p\},\{q\}$.Then by Urysohn's Lemma there exists a continuous function $f:X\to [a,b]$ such that $f(p)=a,f(q)=1$.
As $X$ is connected so $f(X)$ is connected and connected subsets of $[a,b]$ are only $[m,n]$ and $(m,n);a\le m\le n\le b$ which are uncountable and hence $X$ is so.
Is the proof correct?
However I am stuck on the second question ?How should I use the fact that the space is regular ?Any help.
The first proof is quite correct: it's easier even to say that $f[X] = [0,1]$ (just use $a = 0, b= 1$, as in the standard formulation of Urysohn), as the only connected subset of $[0,1]$ that contains $0$ and $1$ equals $[0,1]$ (if it missed $a \in (0,1)$ we'd have an immediate disconnection). And $|X| \ge |f[X]|$ etc.
Suppose $X$ is countable. Then $X$ is Lindelöf (trivially) and a regular Lindelöf space is normal (see this answer, e.g., or search for other proofs if your text does not cover this). Now by the previous part, $X$ is uncountable, contradiction.