Why is inf and sup of null sets defined as infinities

Question

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?

Answer 1

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 $$\inf\varnothing=+\infty,\quad \sup\varnothing=-\infty.$$

Answer 2

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.

Answer 3

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.

Answer 4

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 $+\infty.$