What sets contain $\infty$ and $-\infty$ and why are the Integers closed? So I'm currently studying from Rudin's Principles of mathematical analysis or colloquially "Baby Rudin" and have stumbled into the second chapter namely basic topology. He lists some sets and states whether or not they are bounded, open, closed or perfect. My question com3es from the fact that he calls the set of all integers closed
By the text a set is closed if every limit point is an element of the set itself. Naturally I understand the only limit points of the Integers to be $\infty$ and $-\infty$... however I assumed that the Integers don't contain either of these elements so I reasoned that the Integers were not closed
Could someone explain why this reasoning is wrong? I presume I'm misunderstanding something...
As a corollary question I was wondering which sets contain $\infty$ and/or $-\infty$ 
Continuing the story of my study I then assumed that I was wrong and that the Integers do in fact contain $\infty$ and $-\infty$ ( considering the set of complex and real numbers are as well considered closed I assumed that $\infty$ and $-\infty$ are elements of all these sets ) but then Rudin again talks about the set $S = \left\{\frac{1}{n} | \, n \in \mathbb{N} \right\} $ but says that $0$ is not an element (but is obviously a limit point)... I guess the confusion I have comes from the fact that I then assumed that $\infty$ is an element of the natural numbers and earlier he defines $\frac{1}{\infty} = 0$ so then $0$ should be in the set...
Where is my thinking going awry?
Thanks in advanced!
 A: The integers are closed in $\Bbb R$, the space of real numbers; $\infty$ and $-\infty$ are not in that space and therefore are not relevant. Judging by a quick look at my second edition, he has not at that point talked about $\pm\infty$ or the extended real numbers at all.
A: A set $U$ is neither open nor closed implicitly. In the real line $[0,1)$ is neither open nor closed, but in the interval $[0,1]$ it is open. Whether a set is considered open is relative to a parent topology.
So we say $U$ is open/closed in another space  $X$. Sometimes, the $X$ is implicit, but when somebody tells you a set $U$ is open, there is always either an explicit or implied $X$.
For example, if $U$ is the set of even numbers, you might think "$U$ isn't open." But it is, when considered as a subset of the integers. The even numbers are not open in the set of real numbers, but that is not a contradiction.
The integers, then, are closed as a subset of the real line. As you rightly note, the integers are not closed as a subset of the extended real line, but I doubt Rudin is asserting that, and there is no contradiction there.
There are some general things you can say. If $U\subset V\subset W$ are topological spaces, then if $U$ is open (closed) in $V$ and $V$ is open (resp. closed) in $W$, then $U$ is open (resp. closed) in $W$.
This doesn't help, though, because if $U$ is the integers, $V$ is the real line, and $W$ is the extended real line, we can't conclude anything from $U$ being closed in $V$ because $V$ is not closed in $W$.
A: A limit point in a metric space $X$ of a subset $A\subset X$ is a point $x\in X$ for which a sequence $(a_i)_{i\geq 0}$ converges to $x$, with $a_i\in A$ for all $i\geq 0$. The only convergent sequences of elements of $\mathbb{Z}$ in $\mathbb{R}$ (that is, with the induced topology/metric) are sequences which are eventually constant, further the limit point of such a sequence is simply the integer at which it is constant. It follows that $\mathbb{Z}$ contains all of its limit points and is closed.
A: The set of integers contains no limit points and it follows that it is closed, vacuously so. Rudin briefly discusses the extended real numbers in a small section in chapter one towards the end, however he is not using that system in chapter. I would assume if he were it would be stated explicitly.
A: A "limit point" in topology has nothing to do with the limit of a set.  (Which is why I presume you came up with the idea that $\infty$ and -$\infty$ were limit points of the integers?)  A limit pt of a set $A$ is a point $x$ such that every neighbor of $x$ contains a point (other than $x$) of the set $A$.
So, for instance, the point $\frac{1}{2}$ is a limit point of $[0,1)$ because every neighborhood around $\frac{1}{2}$ contains points other than $\frac{1}{2}$ that are in $[0,1)$.  (Nothing to do with the limits of $[0,1)$.
So a closed set is one which contains all it's limit pts.  Example:  Every neighborhood of $1$ contains some points in $[0,1)$ so $1$ is a limit point of $[0,1)$.  But $1$ is not a member of $[0,1)$ so $[0,1)$ is not closed. 
Consider the integers.  If $x$ is an integer than the neighborhood $(x - \frac{1}{2}, x + \frac{1}{2})$ does not contain any integers other than $x$.  So $x$ is not a limit point of the integers.  Consider $x$ not an integer.  Then you can find a neighborhood around $x$ that has no integers in it.  So $x$ is not a limit point of the integers either.  So the integers have no limit points.  If there are no limit points then it is true that all  of the limit points are in $\mathbb{Z}$ because there aren't any limit points.  So the integers are closed.
So what sets contain $\infty$ and -$\infty$.  Well, as for as Rudin is concerned, only the "extended real number line".  And the only purpose for the extended real number line is to have symbols for unbounded quantities.
