In the following picture
the first two (top) polygons are monotone, whereas the last two (bottom) polygons are not, with respect to the line $L$. At page $51$ of the book "Computational Geometry: Algorithms and Applications" (3rd edition) by Berg et al., the authors state
Since $P$ is not monotone, there is a horizontal line $\ell$ that intersects $P$ in more than one connected component.
It is not clear to me what a connected component is, in this context, even though I am aware of connected components of graphs.
What is a formal definition of a "connected component" in this context? Why "connected"? What is connected? What is an example of a non-connected component? What are the components? Components of what?
Judging by the pictures, what you probably want is for every line to intersect the polygon in a single connected component.
An attempt at informal formality:
Connected subset: A set that cannot be split into two disjoint, nonempty open sets.
Connected component: a connected subset that is maximal, i.e. is not contained in any larger connected subset. This is a largest possible connected chunk, or component, of the space. It turns out that any space splits into a unique collection of connected components. For example: a point is a connected subset of an interval, but all of the points in the interval sit inside the interval, which is connected itself, so the interval is the only connected component of itself.
Why connected: If you can split your subset into several disjoint, nonempty open pieces, those pieces are not connected but just sitting next to each other. Examples are a single point or an interval.
Non-connected component: There is no such definition. For an example of a disconnected subset, consider two separate points on the polygon, or two edges that aren't touching.