# Inverse limit of irreducible spaces

Let $(X_{i})_{i \in \mathbb{N}}$ be an inverse system of topological spaces. Assume that each of the $X_{i}$ is irreducible. Then is it true that $\projlim X_{i}$ is also irreducible?

I read in a paper that the inverse limit of irreducible schemes are irreducible, so this is the motivation for the question.