Term for a general subset that does not include infinity I tried to find the answer to my questions for a while but did not succeed, and I hope that was not only because of deficits in my search terms.
My question is as follows:
Let $\mathcal{A}$ be a set of 3-dimensional positive real numbers, i.e. $\mathcal{R}^{+} \times \mathcal{R}^{+} \times \mathcal{R}^{+}$. I am now looking for a term that describes a subset $\mathcal{B}$ of $\mathcal{A}$ that may include points of $\mathcal{A}$ but not $\infty$. 
The background is that $\int_{\mathcal{B}}1\;dx$ can be interpreted as the (geometric) volume of $\mathcal{B}$ if $\mathcal{B}$ does not include $\infty$.
Is this a closed subset, or a proper subset? As far as I understood both are not correct.
I would like to says something like 
"Define $\mathcal{B}$ as a ... subset of $\mathcal{A}$" 
and this expression shall make clear $\mathcal{B}$ may contain any points in $\mathcal{A}$ but may not include infinity. Or is there no specific expression and shall simply say
"Define $\mathcal{B}$ as a subset of $\mathcal{A}$, where non of its limits is infinity" ?   
Thanks for every suggestion, Johannes
 A: You are slightly confused about notation in your question, but I think I understand what you are asking.
You say that $B\subset A$ may not include infinity, but in reality, since $A$ only contains triplets of real numbers, even the set $A$ does not include infinity. This means that for any set $B\subseteq A$, the statement "$B$ does not include infinity" is perfectly correct.

That said, what I think you are asking is how to describe a set that does not contain arbitrarily large elements. For example, $S=\{(x,x,x)|x>0\}$ contains the numbers $$(1,1,1),(2,2,2),\dots, (1000,1000,1000),\dots, (N,N,N),\dots$$ where $N$ is arbitrarily large. If that is indeed what you are asking, then your terms "closed" and "proper" are incorrect:

*

*A set is closed intuitively, if it contains its own border. This means that $A$ itself is a closed set and, since $A$ obviously contains arbitrarily large numbers, this is not the term you need. Also, the set $S$ described above is also closed.


*A set is a proper subset of $A$ if it is not equal to $A$. This means that $A\setminus\{(1,1,1)\}$ is a proper subset of $A$, but again, it contains arbitrarily large numbers, so "proper" is not the term you need. Again, the set $S$ described above is also closed.
and the term you are looking for is BOUNDED
A: I think you are describing a bounded subset.
