# Why do geometric sets such as $(\infty, x]$ never have infinity included?

I have a question about the use of infinity and geometric sets. Say I am trying to graph an equation, and the result is all values greater than or equal to, say, $3$. From what I've seen, the proper way to write this is $(\infty, 3]$, where $3$ is included, as expected, but infinity isn't. Why?

I realize $\infty$ is obviously infinite, and therefore this is hard to imagine, but wouldn't $\infty$ then be included. Otherwise, it seems like the value would NOT include $\infty$.

I can kind of understand, as $\infty-x$ is still infinity. But therefore, should infinity ALWAYS be included? This is a real head-scractcher.

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The "half-open" interval you want is $[3, +\infty)$ to denote all values greater than or equal to $3$ : $\{x \in \mathbb{R}, x \geq 3\}$.

You can think of the "open" part of the notation indicating that $+ \infty$ (like $-\infty$) has "no bound".

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Ah, thanks. That makes a lot more sense. – Josiah Dec 5 '12 at 15:53
@AceLegend Yeah, that's how I first made sense of the notation! – amWhy Dec 5 '12 at 15:54

$-\infty$ and $+\infty$ are not real numbers. By writing for example $(0,+\infty)$ we are referring to all real numbers greater than $0$. In the extended real number system which is the set of all real numbers with $-\infty$ and $+\infty$ adjoined, we have $-\infty<x<+\infty$ for any real number $x$. This explains the use of the interval notation. If we wrote $(0,+\infty]$ instead we would be including $+\infty$ in our set.

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Hmm. I still don't quite understand. I understand that they aren't real numbers, but why does that mean that they can't be included. I guess my real question here is why $\infty$ can't be included. – Josiah Dec 5 '12 at 15:47
Otherwise, it seems like we are saying all values greater than or equal to 3, but less than $\infty$. – Josiah Dec 5 '12 at 15:47
Really? Well, alright then. I guess that would make sense. – Josiah Dec 5 '12 at 15:52
If you talk about a set including $\infty$ it certainly is no subset of $\mathbb R$. One can talk about extensions of $\mathbb R$ that do include $+\infty$ and $-\infty$ (or in fact just a single signless $\infty$), but be awre that one no longer talks about a subset of a complete ordered field. – Hagen von Eitzen Dec 5 '12 at 16:00
@JasperLoy, I was being sarcastic. Didn't really come out like that though. Either way, I appreciate the answer. Unfortunately, I am not so experienced in mathematics as yourself. Therefore, I could not really make out what your answer was saying. I was not aware that because infinity wasn't in the geometric set that it could not be included. However, amWhy cleared it up for me. – Josiah Dec 5 '12 at 16:07