Why is inf and sup of null sets defined as infinities I'm reading the definition of $inf\emptyset$ and $sup\emptyset$.
a) I'm wondering why $inf\emptyset = \infty$ and $sup\emptyset = -\infty$. I would have expected both to be undefined.
b) In general, can something equal infinity if it's not in the extend real number system? Should I assume they are using about extended real numbers in these definitions?

As others have said, this assumes that we are working in the extended reals.
Contrary to what the others said, though, $\sup\emptyset=-\infty$ is not a definition made out of convenience. It is a direct consequence of the normal definition of the supremum: the smallest upper bound of the set. Since everything is an upper bound of the empty set (everything is larger than all its elements), the smallest upper bound is $-\infty$.
Essentially the same applies to the infimum.
