What does $[0, \infty]$ mean?

Question

Can we "close" the subset with a bracket on the right of infinity like: $[0, \infty]$?

What is the difference to $[0, \infty)$, which is already considered a closed set?

Answer 1

When you "close" the bracket, it implicitly means that you're working in the space $\Bbb R\cup\{\pm\infty\}$ (and we omit the $+$ from the positive infinity sign), and in that space $[0,\infty)$ is not closed, since $\infty$ is indeed a limit point of that set.

[It might be the case that you are working in the $1$-point compatification of $\Bbb R$, which is like "tying" both ends of $\Bbb R$ into a single point denoted by $\infty$, but then the space is not ordered, so talking about intervals becomes a bit awkward.]

Remember that "open" and "close" are relative to a space and to a topology. $\{1\}$ is not open as a subset of $\Bbb R$, but it is open as a subset of $\Bbb N$ (in their standard topologies).

Answer 2

Sometimes, we want to consider $\infty$ as a value that can be attained. For example, we could define $$ f(x) = \begin{cases} 1/|x| & x \neq 0\\ \infty & x = 0 \end{cases} $$ In this case, we would say that $f:\Bbb R \to (0,\infty]$ is an onto function

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We can consider $\Bbb R = (-\infty,\infty)$ as a subset of the larger space $\overline{\Bbb R} = (-\infty,\infty) \cup \{\pm \infty\}$. Under this consideration, we would state that $\Bbb R$ is open (an not closed) in $\overline{\Bbb R}$.

Answer 3

Note that $\infty \notin \mathbb{R}$, so it doesn't make sense to say that $[0, \infty]$ is a subset of $\mathbb{R}$. On the other hand, the set $$[0, \infty) = \{x \in \mathbb{R} \mid 0 \leq x\}$$ completely makes sense and defines a subset of $\mathbb{R}$.

However, one can consider the extended real numbers $\overline{\mathbb{R}} = \mathbb{R}\cup\{-\infty, \infty\}$ and extend the order structure from $\mathbb{R}$ by declaring $-\infty < x < \infty$ for every $x \in \mathbb{R}$. In this setting, the set $$[0, \infty] = \{x \in \overline{\mathbb{R}} \mid 0 \leq x \leq \infty\}$$ is a perfectly well-defined subset of $\overline{\mathbb{R}}$, as is $[0, \infty)$ which can be written as $$[0, \infty) = \{x \in \overline{\mathbb{R}} \mid 0 \leq x < \infty\}.$$

Answer 4

In some cases, it is interesting (or simply makes notation simpler) to consider the extended real numbers: $\overline{\mathbb{R}}=[-\infty,\infty]$.

Formally, pick your favorite objects which are not real numbers, and let's conveniently call them $-\infty$ and $\infty$. Define the set $\overline{\mathbb{R}}=\mathbb{R}\cup\left\{-\infty,\infty\right\}$. Define the following order on $\overline{\mathbb{R}}$: If $a$ and $b$ are real numbers, $a\leq b$ is the usual order in $\mathbb{R}$. For all $a,b\in\overline{\mathbb{R}}$, we also define $-\infty\leq b$ and $a\leq\infty$. This says, intuitively, that $\infty$ is "infinitely large" and $-\infty$ is "infinitely negative" (not small, because this may cause some confusion).

If we were working only with $\mathbb{R}$, we have to specify what the notation $(a,\infty)$ means: the symbol $\infty$ alone does not have any meaning in $\mathbb{R}$. However, in $\overline{\mathbb{R}}$, $\infty$ has a meaning, and the interval $(a,\infty)$ is simply $\left\{b\in\overline{\mathbb{R}}:a<b<\infty\right\}$. Similarly, we may speak of the intervals $(a,\infty]$, $[-\infty,b)$, etc...

The space of extended real numbers has several nice properties. Specificaly, it is isomorphic (in almost every analytical sense) to the interval $[0,1]$, which is certainly one of the nicest spaces out there.