# Maximal connected component.

Let $f: X \to Y$ an application between two topological spaces $\mathbb{X}$ and $\mathbb{Y}$ both Hausdorff. We know that if $Y \in\mathbb{Y}$ is connected and $f$ is continuous then $f^{-1}(Y)$ is connected.

But if $A$ is not connected or if $f$ is not continuous then $f^{-1}(Y)$ is not necessarily connected.

And in this case it is customary to work with the so-called maximal connected component.

Question 1: What is maximal connected component?