Can someone explain these two examples of an open set

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Discussion Overview

The discussion revolves around the concept of open sets in mathematics, specifically focusing on examples of open balls in both real and complex spaces. Participants explore definitions and properties of open sets, including the union of open sets.

Discussion Character

  • Conceptual clarification
  • Technical explanation
  • Meta-discussion

Main Points Raised

  • One participant notes that an open ball in R is proven to be an open set and suggests that the original definition of an open set may stem from this.
  • Another participant extends this logic to open balls in the complex plane, C, indicating a similar reasoning applies.
  • A definition of an open set is provided, stating that a set U is open if for every element x in U, there exists an open set S (possibly an open ball) such that x is in S and S is fully contained in U.
  • There is a mention of a participant resurrecting older threads, indicating a concern about the relevance of the discussion's timing.
  • Another participant acknowledges the date of the thread but expresses a belief that the topic remains open for discussion.
  • A later reply confirms that a staff member is okay with the continuation of the discussion.

Areas of Agreement / Disagreement

Participants appear to agree on the definition of open sets and the properties discussed, but there is a disagreement regarding the relevance of revisiting older threads. The discussion remains somewhat unresolved regarding the appropriateness of continuing the conversation in an older thread.

Contextual Notes

The discussion includes references to specific mathematical concepts and definitions that may depend on the context of real and complex analysis. There are also limitations regarding the accessibility of information based on the device used by participants.

Austin Chang
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4) Apparently they have already proven that the open ball in R is an open set. That may be their original definition of an open set. They have also proven that any union of open sets is open. So all they need to do is to rewrite the original set of 4) as the union of balls.
5) Same logic as in 4) except that the balls are in the complex plane, C.
 
Essentially a set U is open if for every x in U there is an open set ( possibly an open ball) S so that x is an element of S and S is fully contained in U . Can you take it from there @Austin Chang ?
 
WWGD said:
Essentially a set U is open if for every x in U there is an open set ( possibly an open ball) S so that x is an element of S and S is fully contained in U . Can you take it from there @Austin Chang ?

You seem to be going round resurrecting a lot of old threads. This one is from 2016.
 
PeroK said:
You seem to be going round resurrecting a lot of old threads. This one is from 2016.
I didn't notice the dates. I guess they are still open though. I am using my phone, which does not give me the same access as PC.Edit : I will ask staff if it is ok.
 
Last edited:
@PeroK , I just spoke with Greg, he seems to be ok with it. You may chime in if you wish.
 

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