Connected set cannot be written as the union of numerable proper connected separated sets. Is there a proof or a counterexample to such an affirmation? For example, we know that no finite union of proper connected sets is connected. Can it happen that given any connected set, and given a numerable collection of proper connected separated subsets of X, that X is equal to the union of such a collection? In other words, given a connected set $X$, is $X$ is always bigger than any union of proper connected separated sets?

Let $X$ be any countably infinite set, with the cofinite topology.

Then $X$ is connected, and also, $X$ is the countable union of its singleton subsets, which are proper connected sets.

Also, any two distinct singletons sets are separated.