Determining whether or not a set is open

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In summary, the conversation is discussing a problem involving a set in the complex plane and whether or not it is open or closed. The speaker is unsure and asks for clarification. They discuss the boundaries and limit points of the set and come to the conclusion that the set is neither open nor closed. They also mention another problem they are struggling with and ask if the other person is knowledgeable in "epsilon-delta" proofs. It is revealed that the other person is indeed proficient in these proofs.
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The problem specifically refers to a set in the complex plane, and since I'm not sure if that belongs here, I will ask this question generally to make sure I have the right idea.


I was given two equations and graphed the intersection. The intersection looks like a three-quarters of a closed ring, not including the first quadrant. All the boundaries except for one are included in the set.

I was asked whether or not the set is open.

I am lead to believe that since you cannot surround a point on one of the included boundaries with an open disk also included in the set, the set is therefore NOT open. But since the set does not contain all its the limit points, it is also not closed... therefore its neither? I'm getting confused.
 
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  • #2
Hi xokaitt! :wink:
xokaitt said:
I am lead to believe that since you cannot surround a point on one of the included boundaries with an open disk also included in the set, the set is therefore NOT open. But since the set does not contain all its the limit points, it is also not closed... therefore its neither? I'm getting confused.

Yes, some sets are neither open nor closed. :smile:
 
  • #3
this is a different problem that I'm struggling with , but are you competent in delta-epsilon proofs of continuity?
since ur online and i can maybe take advantage if you are :smile: lol
 
  • #4
ill post a new thread i guess.
 
  • #5
And, indeed, there are sets that are both open and closed.

And, yes, tiny-tim is very proficient at "epsilon-delta" proofs!
 

1. What is an open set?

An open set is a set in which all of its elements are contained within an open interval. This means that every point in the set has a neighborhood that is also contained in the set.

2. How do you determine if a set is open?

To determine if a set is open, you can check if every point in the set has a neighborhood that is also contained in the set. This can be done by checking if the set contains any of its boundary points.

3. Can a set be both open and closed?

Yes, a set can be both open and closed. In fact, the only sets that can be both open and closed are the empty set and the entire space itself.

4. What is the difference between an open set and a closed set?

An open set contains all of its boundary points, while a closed set contains its boundary points. Another way to think about it is that an open set does not include its endpoints, while a closed set does include its endpoints.

5. How is the concept of an open set used in mathematics?

The concept of an open set is used in many areas of mathematics, including topology and analysis. It is a fundamental concept that helps to define and understand the properties of various mathematical structures, such as topological spaces and metric spaces.

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