A characterization of the uniform convergence points set of some convergent sequence of functions

We characterize the uniform convergence points set of a pointwisely convergent sequence of real-valued functions defined on a perfectly normal space. We prove that if $X$ is a perfectly normal space which can be covered by a disjoint sequence of dense subsets and $A\subseteq X$, then $A$ is the set of points of the uniform convergence for some convergent sequence $(f_n{)}_{n\in \omega }$ of functions $f_n:X\to \mathbb{R}$ if and only if $A$ is $G_{\delta }$-set which contains all isolated points of $X$. This result generalizes a theorem of J'{a}n Bors'{i}k published in 2019.