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Regions; "Each point of the set is the center of a circle..."by Nathanael
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#1
Aug1914, 08:17 PM

P: 683

"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? 


#2
Aug1914, 09:08 PM

Mentor
P: 3,975

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. 


#3
Aug1914, 09:14 PM

P: 508

http://commons.wikimedia.org/wiki/Fi...__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)##. 


#4
Aug1914, 09:39 PM

P: 683

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


#5
Aug2014, 06:26 AM

Math
Emeritus
Sci Advisor
Thanks
PF Gold
P: 39,693

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.



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