Regions; "Each point of the set is the center of a circle "

[Nathanael]

"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?

 

[Nugatory]

1. Each point of the set is the center of a circle whose entire enterior consists 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?

It says "a circle", not "every circle, no matter how large".
You could read it as saying that if a given point is in the region then there is some distance, perhaps not very large, such that every point closer than that distance to the given point is also in the region.

 

[gopher_p]

http://commons.wikimedia.org/wiki/File:Open_set_-_example.png

$U$ in the picture is a region. It's open (condition 1) and path connected (condition 2). Note the (open) circle around $x$ (denoted $B_\epsilon(x)$, standard notation for "ball of radius $\epsilon$ centered at $x$") which is contained entirely within $U$. The dotted boundaries are meant to indicate that they aren't included as part of $U$ and $B_\epsilon(x)$.

 

[Nathanael]

Thank you, I believe I understand now.

Edited;
Removed what I said because it wasn't what I meant (nor did it make much sense)

 

[HallsofIvy]

The word "enterior" in your initial post confused me. I did not know if you meant "interior" or "exterior"! And your post seemed to indicate that you were confused about that also.

 

[Nathanael]

The word "enterior" in your initial post confused me. I did not know if you meant "interior" or "exterior"! And your post seemed to indicate that you were confused about that also.

Sorry! That was just a typo that I failed to notice. I indeed meant interior.