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!
Best Answer
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.