$X=(-\infty,+\infty]$, $T_>:=\{(-a,+\infty]:a \in [-\infty, +\infty]\}$. Is $X$ compact?

Question

Let $X=(-\infty,+\infty]$ and $T_>:=\{(-a,+\infty]:a \in [-\infty, +\infty]\}$.

$T_>$ is obviously a topology. How can I reason that $X$ is not compact? That is there exists an open cover of $X$ which has no finite subcover.

I thought about $\cup_{n \in \mathbb{N}}(-n,+\infty]$ which is an open cover of $X$ but for every $N \in \mathbb{N}$ $\cup_{n=1}^{N}(-n, +\infty)$ does not cover $(-\infty,N]$ so $\cup_{n \in \mathbb{N}}(-n,+\infty]$ has no finite subcover.

Answer 1

Yes, the cover $(-n, +\infty]$ is a nested cover (the sets grow larger with increasing $n$) and so any finite subset of them has as their union the one with the maximal $n$, and so a finite subset never covers all of $X$.