Let's say we have two continuous random variables $X$ and $Y$. Then the Kolmogorov-Smirnov metric $K(X,Y)$ is defined as \begin{equation} K(X,Y):=\sup_{t}\{|F_{X}(t)-F_{Y}(t)|:t\in\Bbb R\}, \end{equation} where $F_{X}$ and $F_{Y}$ are the distribution functions of $X$ and $Y$, respectively. In every textbook I have looked in, $K$ is always defined in terms of the supremum absolute distance between the two cdfs. That said, it seems that this is (always?) equal to the absolute maximum difference between the two cdfs, i.e., \begin{equation} K(X,Y)=\max_{t}\{|F_{X}(t)-F_{Y}(t)|:t\in\Bbb R\}. \end{equation}
Are both expressions above for $K(X,Y)$ equivalent? If so, why is it not defined in terms of $\max$ instead of $\sup$? If not, could you provide an example where the two give different solutions for $K$?