Open cover, finite subcover Find an open cover of $(-\infty,0]$ that does not have a finite subcover.


Can anyone help me with this? Please explain the intuition of finding it too.

Thank you so much!
A very easy solution is just $$\bigcup_{n=1}^\infty(-n,0].$$ Clearly it can't be finite, and it's open.

If the space isn't to be taken as $(-\infty,0]$, then take the union to be $$\bigcup_{n=1}^\infty(-n,1).$$
Maybe you want $C = \{(-n-2, -n+2), n \in \mathbb{N}\}$

Let $V_n = (-n-2, -n+2)$

Then $[0, \infty) \subseteq \bigcup\limits_{n = 1}^\infty V_n $

Try to find an finite subcover $U$ such that $[0, \infty)  \subseteq U$, you can't!