On definition 2.68, the book states that a set in the plane is called a region if it meets two conditions (p. 14):
"Each point of the set is the center of a circle whose entire interior consists of points of the set."
"Every two points of the set can be joined by a curve which consists entirely of points of the set."
I have been trying to understand what the first part of condition really means and I am not sure if I am grasping the concept of the first condition. Can anyone help decipher its meaning? Can anyone provide a visual example?
Thank you,
$1$ refers to "openness" and $2$ refers to "connectedness".
What $1$ means is that if you choose any point in the set, you can draw an entire circle around it, with the disk (which is the circle and the region bounded by it) being completely inside the set.
Example:
The first quadrant, excluding the axes, is a region. But if the axes were included, it would not be a region.