"A set in the plane is called a region if it satisfies the following two conditions: 1. Each point of the set is the center of a circle whose entire enterior consists of points of the set. 2. Every two points of the set can be joined by a curve which consists entirely of points of the set." I'm having trouble understanding the meaning of the first condition. Can someone please try to explain it in different words? The way I'm understanding it, it seems to say that only an entire plane can be a region. (But this is obviously incorrect?) How does the first condition allow for a bounded region?