Thread: Is a compact space closed?

1. Is a compact space closed?

Every compact set is closed. So the union of all compact sets is closed? If that's so, then a compact space, which is basically the union of the compact sets, is closed, right?

2. Hi Mohit.

Originally Posted by Mohit
So the union of all compact sets is closed?
No. The union of finitely many compact sets only is closed.

3. Originally Posted by TheAbstractionist
Hi Mohit.

No. The union of finitely many compact sets only is closed.
Yes, this is correct. For an example consider the following. $\displaystyle K_n=[-n,n]$, each of which is clearly compact, now $\displaystyle \bigcup_{n=0}^\infty K_n=\mathbb{R}$, which is not compact (it is not bounded).

4. Originally Posted by Mohit
Every compact set is closed (not necessarily true). So the union of all compact sets is closed? If that's so, then a compact space, which is basically the union of the compact sets, is closed, right?
If your topological space is a standard topology on $\displaystyle \mathbb{R}$, then a subset A of $\displaystyle \mathbb{R}$ is compact if and only if it is closed and bounded.
More generally, if a topological space X is a compact Hausdorff space, then a subset A of X is compact if and only if it is closed. In this case, as Abstractionist mentioned, finite unions of compact sets is compact.

However, open sets can be compact sets. For example, in a cofinite topology on $\displaystyle \mathbb{R}$, compact sets are not necessarily closed sets.

5. Thank you all, thanks a lot.

6. Originally Posted by Mohit