# Totally disconnected locally compact Hausdorff spaces

Can any totally disconnected locally compact Hausdorff space be written as a disjoint union of subsets that are both compact and open?

If this is true, does anyone know of a good reference?

• If you have a group, then yes. – Marc Palm Dec 30 '13 at 11:09

The set $\omega_1$ of countable ordinals with the order topology is a totally disconnected locally compact Hausdorff space which can not be written as a disjoint union of subsets that are both compact and open. This follows from the fact that the space is sequentially compact but not compact.
Every paracompact locally compact space can be partitioned into a collection of $\sigma$-compact open sets (for a proof of this result see my answer to another question here).
Suppose that $X$ is a locally compact paracompact totally disconnected space. Then $X$ can be written as a disjoint union $\bigcup_{i\in I}X_{i}$ of $\sigma$-compact open sets. Since each $X_{i}$ is $\sigma$-compact, the sets $X_{i}$ can be written as countable unions $\bigcup_{n}C_{i,n}$ of compact clopen sets. If $D_{i,n}=C_{i,n}\setminus\bigcup_{m<n}C_{i,m}$, then $D_{i,m}$ is a partition of $X_{i}$ into compact clopen sets. Therefore $\{D_{i,n}|i\in I,n\in\mathbb{N}\}$ is a partition of $X$ into compact clopen sets.