How to determine if a set is an open subset of a Euclidean space?

  • Thread starter kelp
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  • #1
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I opted to not use the template because this is a pretty general question. I am not understanding how to find out if a set is an open subset of a Euclidean space.
For example,
{(x,y) belongs R2 | x squared + y squared < 1}
The textbook is talking about open balls, greatly confusing me.
 

Answers and Replies

  • #2
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Intuitively speaking, a set is open if you can take any point in the set, move a little distance in any direction and still be in the set. In this situation, the set of points (x,y) such that x^2 + y^2 < 1 is really the "inside" of a circle of radius one (specifically we are dealing with the unit circle). If your point is near the center of the circle, it is obvious that you can move some distance away and still remain inside the circle. But the idea is that if we have a point near the "edge" or the boundary of the circle, we can still move closer and closer towards the edge without hitting the boundary, and this is why we have the inequality x^2 + y^2 < 1 (when < is replaced with =, we are on the unit circle). Does this make sense?

Hmm so I guess I gave away the answer but not really. Do you need to prove the set is open?
 

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