The following is a paragraph from "Digital image processing, 4th edition, Gonzalez and Woods".
Let S represent a subset of pixels in an image. Two pixels p and q are said to be connected in S if there exists a path between them consisting entirely of pixels in S. For any pixel p in S, the set of pixels that are connected to it in S is called a connected component of S. If it only has one component, and that component is connected, then S is called a connected set.
I do not understand the last sentence: How can S have one component and that component is not connected??
Another question: In a binary image, can S be a connected component and have both 0 and 1 pixels?
By the way, the concept of component is not define precisely in the book.
In the book, a region is defined as a connected set. The following figure is an example of two adjacent regions (So, $R_i$ and $R_j$ are both connected sets.)