# Compactness and Open Sets in $\mathbb{R}^{d}$

I read the appendix of Support Vector Machines book on the other day and wondered about how we can show that an open set in $\mathbb{R}^{d}$ is not compact. Let me start by giving the definition of compactness and a related theorem. These are from the book.

Definition A set $A$ in a topological space is compact if for every family $(O_{i})_{i\in I}$ of open sets with $A\subset\cup_{i\in I}O_{i}$, there exist finitely many indexes $i_{1},\ldots,i_{n}\in I$ with $A\subset\cup_{j=1}^{n}O_{i_{j}}$. In other words, $A$ is compact if each of its open covers has a finite subcover (from this Wiki page).

Theorem $A\subset\mathbb{R}^{d}$ is compact if and only if $A$ is closed and bounded.

Here is what I wondered after reading these two items. Consider an open unit sphere $S=\left\{ x\mid\|x\|<1\right\}$ in $\mathbb{R}^{d}$. Obviously by the theorem $S$ is not compact because it is not closed. That means there must exist an open cover such that there is no finite subcover. What kind of open cover is that ? That is my question.

I got my answer after discussing with Zoltan. What we need are two things. Firstly, we need to find open covers $\{O_{i}\}_{i=1}^{\infty}$ that can cover $S$ i.e., $S\subseteq\cup_{i=1}^{\infty}O_{i}$. Secondly, show that no finite number of $\{O_{i}\}_{i}$ can cover $S$.

Define $O_{i}=S_{r_{i}}$ where $S_{r}=\left\{ x\mid\|x\|<r\right\}$ and $(r_{i})_{i}$ is a monotonically increasing sequence converging to 1 with $r_{i}<1$ for all $i$. No finite number of $\{O_{i}\}_{i}$ will cover $S$ because $\cup_{j\in J}O_{j}=S_{\max\left(\left\{ r_{j}\mid j\in J\right\} \right)}$ for any finite index set $J$ and $S_{\max\left(\left\{ r_{j}\mid j\in J\right\} \right)}\subsetneq S$. To show $S\subseteq\cup_{i=1}^{\infty}O_{i}$, we need to show that for all $p\in S,p\in\cup_{i=1}^{\infty}O_{i}$. This is equivalent to: for all $p\in S$, there always exists an open set $O_{R}\ni p$ for some $R$. The latter statement holds since the open cover sphere can grow to any radius $<1$.