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?

Having$$\inf\emptyset=\infty\text{ and }\sup\emptyset=-\infty\tag1$$is that only way of defining $\inf\emptyset$ and $\sup\emptyset$ so that you always have$$A\subset B\implies \inf A\geqslant\inf B\quad\text{and}\quad\sup A\leqslant\sup B.$$And, yes, you can only have $(1)$ if we are working with the extended real numbers.
