Determining whether or not a set is open

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Discussion Overview

The discussion revolves around determining whether a specific set in the complex plane is open or closed, based on its graphical representation and properties related to boundaries and limit points. Participants explore the definitions and characteristics of open and closed sets, as well as related concepts such as continuity and delta-epsilon proofs.

Discussion Character

  • Debate/contested, Conceptual clarification

Main Points Raised

  • One participant describes a set that appears as three-quarters of a closed ring, questioning whether it is open or closed based on its boundaries and limit points.
  • Another participant agrees that the set cannot be open if it includes boundary points that cannot be surrounded by an open disk entirely within the set.
  • Some participants note that it is possible for sets to be neither open nor closed, indicating a complexity in categorizing the discussed set.
  • A different participant introduces a separate topic regarding delta-epsilon proofs of continuity, suggesting a shift in focus from the original question.
  • Another participant mentions that there are sets that can be both open and closed, adding to the complexity of the discussion.

Areas of Agreement / Disagreement

Participants generally agree that the discussed set is neither open nor closed, but there is no consensus on the implications or definitions involved. Additionally, a separate topic regarding continuity proofs introduces further divergence in focus.

Contextual Notes

The discussion includes assumptions about the definitions of open and closed sets, as well as the properties of limit points, which may not be fully articulated. The introduction of delta-epsilon proofs also indicates a potential shift in the scope of the discussion.

xokaitt
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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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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:
 
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
 
ill post a new thread i guess.
 
And, indeed, there are sets that are both open and closed.

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

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