Compactness. Examples

• The collection of sets $\{(n-1,n+1):n\in \mathbb{Z}\}$ form an open cover of $\mathbb{R}$. This cover has no finite subcover. So $\mathbb{R}$ is not compact.
• Divide the interval $[0,1]$ like so: $U_0=[0,\frac{1}{2}]$, $U_n=(2^{-n},1]$ for $n\ge 1$, then form the cover $(U_n{)}_{n\ge 0}$. This is an open cover, and has many finite subcovers (e.g. ([0, 1/2), (1/2, 1], (1/4, 1]). So we can say that $[0,1]$ is .We can't say anything. We would need to prove that all open covers have a finite subcover to prove compactness.
• The compact subspaces of ${\mathbb{R}}^n$ are the closed bounded subsets.
• Any indiscrete space is compact.
• Any finite space is compact.
• A discrete space is compact iff it is finite.
• In a normed vector space $V$, the closed unit ball is compact iff $V$ is finite-dimensional.