Let $X=(0,\infty)$ and consider the open cover of $X$ $$\mathcal{U}=\{U_1,U_2,U_3,...\}$$ with $U_n=(0,n)$
Q: Show that $\mathcal{U}$ does not admit a subcover which is locally finite.
I find it really hard to show something admits a locally finite subcover or not. Hope someone can help me with this!
Note each of those sets contains $\frac{1}{2}$. Thus, the subcover can be locally finite (which implies that it has e.g. $\frac{1}{2}$ covered with finitely many sets) only if it is finite. But then, it cannot cover the whole set $(0,\infty)$.