Interval Notation and Infinity- Closed or Open?

Question

I was wondering whether I should use closed $[-\infty, \infty ]$ or open $(-\infty, \infty )$ notation when representing the infinity sign in interval notation. My teacher says to use open symbols because infinity has no end, but I was also taught that the open signs $(-\infty, \infty )$ are equivelant to $\lt$ and $\gt$ respectively, meaning it would have a value less than infinity...?

Answer 1

Basically, it is just a question of notation. We agree that $(a,b)$ is defined as the set of all real numbers $x$ for which $a<x<b$.

In this notation, there is no way to describe the set of all values $x$ for which $a<x$. Because we want a notation for this, we agree that this set is the set between $a$ and "infinity". Note here that "infinity" is NOT a real number, so writing $(a,\infty)$ or $(a, \infty]$ is a NEW notation that does not fit into the category of $(a,b)$ or $(a,b]$. This means we can decide to use any of the two notations, however, we use the first.

WHY? The common sense is that $\infty$ is not a number, and writing closed notation would (incorrectly) imply that $\infty$ is an element of $(a,\infty]$. We choose the open notation also because in a sense, we understand $\infty$ to be "something" that is larger than any real number, so in a sense, we think that $\infty>x$ for all $x$.

Answer 2

This seems to be a matter of taste. Consider $f(x)=\frac{1}{1+x²}$. Then $f(0)=1$ and $lim_{|x|→∞}f(x)=0$, so the range of $f(x)$ is a closed interval, as is the case for the image of any finite closed interval. To keep things on the same footing this suggests the notation $[-∞,+∞]$ but I can imagine situations where the open interval notation $(-∞,+∞)$ would be more natural.