Let $X$ be a topological space which, for my intents, may be assumed to be metrizable and compact if needed (let's say it's a closed subset of the unit cube or something like that).
I know that:
- If $X$ has only a finite number of connected components than each connected component is clopen. Generally, they are only closed.
- Each clopen set of $X$ is a union of connected components.
This means that if the Boolean algebra generated by connected components is finite, then the Boolean algebra of clopen sets is finite and the two algebras are equal.
However, if I know that the algebra of clopen sets is finite, can I get that the algebra generated by connected components is finite?
(It is possible to use the fact that in a compact Hausdorff space, connected components and quasi-connected components are the same, but I haven't been able to tie the ends together...)