Open Cover of Compact Set Minus a Point on the Boundary I am having a hard time thinking of an infinite (uncountable or not) open cover of a compact set missing a point on its boundary in $\Bbb R^2$, so that the open cover has no finite subcover. I know this must be possible as the set is no longer closed and thus no longer compact. For example can somebody give me a cover of $$E = \\{(x,y)\in \Bbb R^2 : x^2+y^2\le1 \\} \setminus \\{(0,1)\\}$$ that has no finite subcover.

HINT: For any $p\in\Bbb R^2$ and any $\epsilon>0$, the set $\Bbb R^2\setminus\operatorname{cl}B(p,\epsilon)$ is open. (Here $B(p,\epsilon)$ is the open ball of radius $\epsilon$ and centre $p$.)

**Added:** This doesn’t arise in your specific example, but in general you need more than that $p$ is in the boundary of the compact set: you need it to be a limit point. If your original compact set were the closed unit disk together with a point $p$ not contained in it, for example, $p$ would be in its boundary, but removing $p$ would still leave a compact set.