4
$\begingroup$

I am trying to understand if there is an error in this question, or if the answer is trivial.

Define a topology on $[0,\infty)$ with open sets

  • $(a,\infty), a \in (0,\infty)$

  • $[0,\infty)$ and $\emptyset$.

Show $[0,\infty)$ is compact in this topology.

Since we can only build an open cover out of open sets, surely we can only take $\cup_{n > 0} (n,\infty)$ which does not cover $[0,\infty)$.

Can we either:

  1. Trivially take $[0,\infty)$ as an open finite cover, meaning whenever we take the whole space of any topology it is automatically compact.

  2. The question should read $[a,\infty), a \in [0,\infty)$ are open in this topology

Any help would be greatly appreciated.

Many thanks,

Ash

  • 0
    What Henno means is that you should write $a\in [0,\infty)$ in your first bullet.2011-03-16

1 Answers 1

5

The first variant is correct one. The only possible cover of $[0,\infty)$ in this topology is a cover which has $[0,\infty)$ in it. This follows since $\cup_{a}(a,\infty)=(0,\infty)$, so it is not possible to construct cover of $[0,\infty)$ using only sets $(a,\infty)$ with $a>0$.

  • 0
    @Rasmus, thanks. In my native language there are no articles, so although I know rules about them, I tend to forget how to use them :)2011-03-16