# Compact spaces with atomless algebra of clopen sets

Let $$X$$ be a compact (not necessarily metrizable) topological space. Let $$\mathcal{B}$$ be its algebra of clopen sets and let us assume that $$\mathcal{B}$$ is atomless, i.e. every non-empty clopen subset of $$X$$ can be partitioned into two disjoint non-empty clopen subsets.

This happens e.g. when $$X$$ is totally disconnected. In that case, $$X$$ is an inverse limit of a system of finite spaces. However, assume that $$X$$ is not totally disconnected. Can $$X$$ be expressed as an inverse limit of compacts with finitely many connected (necessarily clopen) components?

• Do you mean to assume all your spaces are Hausdorff? (Or is that included in "compact" for you?) Mar 24 '20 at 14:53
• In particular, your assertion that $X$ is an inverse limit of a system of finite spaces when $X$ is totally disconnected is wrong without that assumption. Mar 24 '20 at 14:59
• It's also not true that $\mathcal{B}$ is always atomless when $X$ is a totally disconnected compact Hausdorff space. (For instance, consider a 1-point compactification of a discrete space.) Mar 24 '20 at 15:02

## 1 Answer

The assumption that $$\mathcal{B}$$ is atomless is totally irrelevant here. Every compact Hausdorff space $$X$$ is an inverse limit of compact Hausdorff spaces with finitely many components. To prove this, first note that we may assume $$X$$ is a subspace of $$[0,1]^I$$ for some $$I$$. Say a subset $$\prod_{i\in I}U_i\subseteq[0,1]^I$$ is a box if each $$U_i$$ is an open interval and $$U_i=[0,1]$$ for all but finitely many $$i$$. Every open set in $$[0,1]^I$$ a union of boxes, so every closed set (in particular, $$X$$) is an intersection of complements of boxes. Moreover, it is easy to see that a finite intersection of complements of boxes has finitely many connected components. So now consider the inverse system consisting of all finite intersections of complements of boxes in $$[0,1]^I$$ that contain $$X$$ (and their inclusion maps). This is an inverse system of compact Hausdorff spaces with finitely many connected component whose limit is $$X$$.