What is the terminology for half interval?
All $x$ where $a \leq x \leq b$, i.e. $[a, b]$ is an interval, or “range“, commonly named.
But all $x$ where $x \geq a$, “i.e. $[a, \infty]$” is not called an open interval – it is a term for all $x$ where $a \lt x \lt b$, i.e. $(a, b)$, or half-open interval for $[a, b)$, or for $(a, b]$.
So, how is the infinite “set” of numbers on a half-line of the number line named? I'd guess e.g. “half interval”, but I don't see it to be in use. (And I'd be more happy for a “trivial” plain English name, like “range” for interval, than e.g. complicated an “interval with an infinite endpoint”.)
UPDATE
Simple English
Let's say I'm talking about natural numbers.
1 to 5, or 1, 2, 3, 4, 5 this is an interval, or range (1–5).
Anything greater or equal 7, or 7, 8, 9, … (to infinity), this is what?
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$\begingroup$$\infty$ is not a member of the standard real numbers so $[0,\infty]$ is not usually used as it would suggest that $\infty$ was included. $[0, \infty)$ and $(0, \infty)$ are commonly used but they are really just suggestive shorthands for set such as $\{x \in \mathbb{R} : x \ge 0 \}$.
$\endgroup$ $\begingroup$The reason why we call an interval 'open' or 'closed' is because such intervals are open, closed, respectively, in the usual topology of $\mathbb{R}$. Since a topological space can also have clopen sets (that is, sets which are both open and closed) and sets which are neither open nor closed, some intervals are not called open/closed because they are in one of these last two cases.
$\endgroup$ $\begingroup$When talking about real numbers, $[0,\infty)$ is an interval, which might be called "unbounded" or "infinite". Because it notationally looks like $[0,1)$, some might call it a "half-open" interval. Also, because it is the complement of the open interval $(-\infty,0)$, those familiar with topology might call it a "closed" interval even though it is not written with two rectangular brackets (since $\infty$ is not a real number).
When talking about integers only, it is not common to refer to "intervals", and especially uncommon to use interval notation like $[3,5]$. But in a discussion of something like order theory, you might use "interval" to mean "any subset $S$ of the ordered set I'm looking at (e.g. the integers) with the property that if $a$ and $b$ are in $S$ and $a<x<b$ (in the original ordered set) then $x$ is in $S$ as well". Then things like "the set of integers greater than $5$" would be an "interval of integers".
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