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?
 A: 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.
A: 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.
A: You can assume that the author used the extended real number line for these definitions.
In fact, here's a motivation for above definition.
If you have two sets $A\subseteq B\subseteq\mathbb R$, then you want them to satisfy
$$\inf A\geq \inf B,\quad \sup A\leq \sup B.$$
You can check that this always works whenever both $A$ and $B$ are non-empty.
We want this to remain true even if we accept $A=\varnothing$. We then must have
$$\inf\varnothing\geq \inf B,\quad \sup\varnothing\leq \sup B$$
for any set $B\subseteq\mathbb R$.
Since you can then choose $B=\{x\}$ for $x\in\mathbb R$ arbitrarily large (or small), we are forced to define
$$\inf\varnothing=+\infty,\quad \sup\varnothing=-\infty.$$
A: You should assume the presence of the "extended real number system" when thinking about the infimum (about the supremum) of a set $M \subset \mathbb{R}$.
You should do this because it is helpful. I visualize it dynamically and non-rigorously, as follows. The extended real number system is a train track or subway line with a western terminus at $-\infty$ and an eastern terminus at $+\infty$.
While traveling from the western to the eastern end, each real number is passed.
Now the algorithm for $\inf$ is as follows. Let the train begin at $-\infty$, and for a fixed set $M$ let a flag be placed at each real $a \in M$.
When the train encounters or hits the first flag, it halts and declares its output as the real corresponding to that first flag. (Why is this not rigorously correct? Consider $\{a \in \mathbb{R}: 0<a<1\}$.)
But, in the case $M=\emptyset,$ the eastbound train never encounters a flag, and proceeds to the end of the line, which is $+\infty$.
The case for $\sup$ is similar but the train is westbound departing from $+\infty.$
