As far as I know, the standard definition of an open set is that the set $A$ is called open if $A \subseteq X$ for some set X and if $A \cap \partial A=\emptyset$ where $\partial A$ is the set of boundary points of A. In particular, I fail to see the motivation for the $A \cap \partial A=\emptyset$ part of this definition, why wouldn't replacing this with $\partial A=\emptyset$ be satisfactory?
Admittedly, experience is telling me that this is a case of "we define it this way because it is useful", but if that is the case then in what way is this useful? After all, as far as I can tell the alternative that I've proposed is very nearly equivalent to the standard definition. The only difference that comes immediately to my mind is that the standard definition would treat the case of $A=\emptyset$ differently. Is that an important difference? Or have I missed something else that is important?