I am studying complex integration on https://www.amazon.com/Complex-Analysis-Springer-Undergraduate-Mathematics/dp/1852337338/ref=sr_1_12?s=books&ie=UTF8&qid=1537368776&sr=1-12&keywords=Complex+analysis
The author states that the Heine-Borel theorem is needed for some proofs later and then explain the concept of open covering of $S \subset \mathbb C$ as a "possibly infinite collection of open sets $V_i$ whose union contains S".
Then the Heine-Borel theorem: "let $S$ be a closed, bounded subset of $\mathbb C$. Then every open covering of $S$ contains a finite subcovering of $S$."
I can follow the proof, but my intuition clashes with it. Why does S need to be closed? The only reason i can come with is if the definition of covering is incomplete and the $V_i$ need to be subset of $S$. Is that so?
Best Answer
It has to be closed because otherwise the statement is false. If $S$ is not closed and if $z\in\overline S\setminus S$, consider the set$$\left\{\left\{w\in\mathbb{C}\,\middle|\,\lvert w-z\rvert>\frac1n\right\}\,\middle|\,n\in\mathbb{N}\right\}.$$It is an open cover of $S$, but it you can check that it has no finite subcover.