I have a question about the proof of Lemma 5.13 in John Lee's text, the proof is shown below.
My question is about the phrase underlined in red, why is a discrete subset of the compact set finite?
Now, the author never gives a definition for discrete subset, he only gives it for discrete topology but from the phrase underlined in blue, I can tell that his definition of discrete set is as follows,
$S$ is a discrete subset of a topological space $X$ if for each $ s \in S$, there exist a neighborhood $U$ of $X$ such that $ U \cap S = \{s\}$. Is this correct?
Then, if you take $X$ to be the closed interval $ [-1,1]$ with the subspace topology induced from $\mathbb{R}$, then $X$ is compact, and let $S = \{ {1 \over n}, n \in \mathbb{N} \} $, then according to the definition above, $S$ is discrete but $S$ is not finite??
I feel I'm missing something but I don't know what it is.
Thank you.
Best Answer
You're right -- this was a mistake in the statement of Theorem 5.13. There are some corrections for this in my online errata list, which you will probably want to download and have handy when you read the book.