• 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/a finite subcover?].
• 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 [...].
• The compact subspaces of $$\mathbb{R}^n$$ are the [...].
• In a normed vector space $$V$$, the closed unit ball is [compact/not compact, possibly with conditions].