Given a manifold which Hausdorff, second-countable. Does there exist a countable open cover $\{U_i\}_{i=1}^\infty$ such that the closure $\bar{U_i}$ of each $U_i$ is compact?
2026-04-11 12:56:06.1775912166
Can every manifold be covered by compact sets?
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It can be done with $\Bbb R^n$, and you can cover your manifold with countably many open sets homeomorphic to $\Bbb R^n$. What can you conclude from this?