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

In summary, a region in the plane is defined by two conditions: 1. Each point is the center of a circle that is entirely contained within the region, and 2. Any two points in the region can be connected by a curve that is also contained within the region. This allows for bounded regions, as the first condition only requires a circle around the point, not the entire plane.
  • #1
Nathanael
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"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?
 
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  • #2
Nathanael said:
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.
 
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  • #3
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)##.
 
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  • #4
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)
 
Last edited:
  • #5
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.
 
  • #6
HallsofIvy said:
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.
 

What does it mean when each point of the set is the center of a circle?

This means that the set is made up of a group of points that are all equidistant from each other, forming a circle around each point.

What is the significance of having each point be the center of a circle?

This property allows for a clear and defined boundary around each point, making it easier to analyze and understand the set as a whole.

How does this relate to regions in mathematics?

In mathematics, a region is a defined area or space. The fact that each point in this set is the center of a circle creates distinct regions within the set.

Can this concept be applied to real-world scenarios?

Yes, this concept is often used in mapping and geographic analysis. For example, a set of cities could be represented by points on a map, with each point being the center of a circle representing the city's influence or radius.

What other properties or characteristics can be observed in a set where each point is the center of a circle?

Other properties that can be observed include the distance between points, the size and shape of each circle, and the overall symmetry of the set.

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