Let $ f: X \to Y $ be an application between two topological spaces $ \mathbb{X} $ and $ \mathbb{Y} $ both Hausdorff. 

The set $f^{-1}(Y)$ is not necessarily connected even if $Y\subset \mathbb{Y}$ is connected. And in this case it is customary to work with the so-called maximal connected component.

Question 1: What is a *maximal connected component* of $f^{-1}(Y)$?

And after Azarel's Answer, I have the following question:

Question 2: What is the maximal connected component of $f^{-1}(y)$ with $y\in\mathbb{Y}$?