Is $[0,\infty )$ a closed interval or a half-closed interval?

Question

Is $[0,\infty)$ a closed interval or a half-closed interval?

My confusion is whether infinity is included or not.

Answer 1

This is a closed set wrt. euclidean metric. Namely, if $x_n\ge 0$ and $x_n\to x$, then $x\ge 0$, which proves the closedness.

Answer 2

$ \infty $ is not included !

By definition: $[0, \infty)=\{x \in \mathbb R: x \ge 0\}.$

$[0, \infty)$ is closed, as SZW1710 has outlined.

Answer 3

It is closed.

Usually, one defines a closed set $B$ as a set whose complement $B^c:=X\setminus B$ is open. Regardless, $[0,\infty)$ is closed iff its complement is open. In this case, it may help to write it like this:

$[0,\infty)^c=\{\,x\in\mathbb{R}\,\mid x\geq 0\,\}^c=\{\,x\in\mathbb{R}\,\mid x<0\,\}=(-\infty,0)$

(Since by taking the complement we are negating the statement in the set builder.) This is clearly an open set, so $[0,\infty)$ was closed. The answer given by szw1710 is also good, since if a set contains all its limit points, it is closed.

The confusion is of course rooted in the feeling that we should have $[0,\infty]$ for it to be a closed set, otherwise it seems it would only be half-closed. This is probably what you mean by 'whether infinity is included or not'. Now, $\infty$ is not a real number, so it would not make sense to talk about $[0,\infty]$ unless you were talking about the extended real numbers. But, even so, as has hopefully been demonstrated, $[0,\infty)$ is closed, and of course so is $[a,\infty)$ or $(-\infty,a]$ for any real number $a$. What we would call half-closed (half-open) is something like $[3,5)$, whose complement is $(-\infty,3)\cup [5,\infty)$, which is a union of one open and one closed interval. (There are also sets which are both open and closed, such as the whole space $\mathbb{R}$, these are called clopen.)