Here's Dini's theorem: If monotone sequence of continuous functions converges pointwise on a compact set $X$ and the limit function is also continuous, then the convergence is uniform.
I wonder if the compactness condition in Dini's theorem can be replaced by its dense set. For example, suppose that there is a set $Y$ dense in a compact set $X$. Can we replace $X$ by $Y$ in Dini's theorem?
$Y=[0,1)$ is dense in the compact set $X=[0,1]$, and the sequence of continuous functions $f_n(x) = x^n$ converges monotonically and pointwise on $Y$ to the continuous function 0, but not uniformly.