I am reading an article about uniqueness of some evolution PDE and at some point, the author says that to show uniqueness on an interval $[0,T]$, by a "classical connectedness argument" (his words) it was enough to show it on a time interval $[0,T'] \subset [0,T]$, for $T'$ arbitrarily small. I am not sure to understand what he means by this.
I thought maybe consider $\{t~|~u_1(t) = u_2(t)\}$, for $u_1, u_2$ two solutions, which is a closed set and then try to show it is open as well so that this gives us uniqueness by connectedness. However I do not see exactly how to do it with the uniqueness on $[0,T']$ arbitrarily small. Does anyone know how to do that ?