Using Urysohn's Lemma, it can be shown that a connected normal space $X$ (with more than one point) is uncountable. But then how can it be that a connected normal space might just be a single point? Is this immediate from Urysohn's Lemma?
No, it’s immediate from the definitions of connectedness and normality. A one-point space is clearly not the union of two disjoint non-empty sets, open or otherwise, so it’s connected. A one-point space doesn’t contain two disjoint closed sets, so the defining condition of normality is vacuously satisfied.