Is the union of a countable number of disjoint compact sets never compact?

Question

As above, I have read this question and it got me thinking. I have not yet been able to come up with a counterexample but I also don't know how to proceed in a way that convinces me there is no case in which the result is compact.

I don't need a proof, just something to kickstart my intution. Thanks!

Answer 1

Α counterexample is $A_n=\{\frac{1}{n}\}$ and $A_0=\{0\}$ on the real line with the usual topology.

The union is compact.

Answer 2

If $A_i, i\in I$ is an infinite family of nonempty topological spaces, then their disjoint union / direct sum (in the sense as defined in https://en.wikipedia.org/wiki/Disjoint_union_(topology) ) is never compact.

Namely, every $A_i$ can be viewed as an open subset of the disjoint union, and those $A_i$ therefore make an open cover of the union which doesn't have a finite subcover.