On p.52 of Mumford's book Algebraic Geometry: Complex projective varieties, he states that $$\mathcal{O}_{x.X} \text{is integrally closed} \ \Rightarrow X \ \text{is topologically unibranch at } \ x.$$ is true but hard to prove. Can anyone supply a proof or references to one ?

Topologically unibranch means there is a neighbourhood basis $\{U_n\}$ for $x$ such that for any algebraic set $Y$, $U_n-Y$ is connected. In fact I have not found this notion anywhere else but in Mumford's book.