If $A\subseteq\mathbb{R}$ is not closed. How can I build an open cover of $A$ such that it does not have a finite subcover?
2026-04-03 13:25:00.1775222700
Open cover of $A\subseteq\mathbb{R}$ such that it does not have a finite subcover.
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HINT: Let $p\in(\operatorname{cl}A)\setminus A$, and find a family of open sets $U_n$ such that $\bigcup_nU_n=\Bbb R\setminus\{p\}$; I’ve suggested a specific one in the spoiler-protected block below. Show that no finite subfamily can cover $A$.