Boundaries of upper semicontinuous set valued maps

Let $x_0$ be a q-point of a regular space $X,Y$ a Hausdorff space whose relatively countably compact subsets are relatively compact and let $F:X\to Y$ be an upper semicontinuous set valued map. Then the active boundary $\mathrm{Frac}F(x_0)$ is the smallest compact kernel of $F$ at $x_0$.