Understanding R^2 Open and Closed Sets

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Discussion Overview

The discussion revolves around the properties of the set R^2, specifically addressing how it can be classified as both open and closed. Participants explore definitions of open and closed sets within the context of topology and analysis, referencing concepts such as limit points and open balls.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant questions how R^2 can be both open and closed, seeking clarification on the definitions of open and closed sets.
  • Another participant explains that R^2 is open because every point is an interior point, and it is closed because it contains all of its limit points.
  • A later reply acknowledges the complexity of the topic, suggesting that the discussion may extend beyond the scope of the participant's current studies.
  • One participant challenges the initial definitions provided, specifically questioning the phrase "less than zero" in the context of open sets, suggesting it may have been a miscommunication.
  • Another participant clarifies that for R^2, any open ball around a point is indeed a subset of R^2, reinforcing the idea of openness.
  • There is mention of the empty set being open, which is used to argue that R^2, as the complement of the empty set, is closed.

Areas of Agreement / Disagreement

Participants express differing views on the definitions and implications of open and closed sets. While some points are clarified, there remains uncertainty and no consensus on the initial definitions and their interpretations.

Contextual Notes

There are unresolved issues regarding the definitions of open and closed sets, particularly the phrasing used by participants. The discussion also highlights a potential misunderstanding of the relationship between open balls and the set R^2.

romsofia
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If we define a set, c, to be R^2, how is it open and closed?

The definitions I'm using:

Open set: Open set, O, is an open set if for all points x are in O, and we can find ONE B(x,ρ) such that B(x,ρ) is less than zero.

Closed set: Compliment of an open set, AKA R^n/O.

This isn't a HW question, I'm reviewing the analysis part of my PDE course from fall semester, and this is something extra she told us, and now I don't know how this could be true!

Thanks for the help.
 
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"Openness" and "closedness" aren't mutually exclusive, despite their unfortunate names. This is obvious topologically (the whole space is open by definition, but it is also the complement of the (open) empty set, and so it is also closed), but there's no need to abstract as far as topology with Rn; that every point in R2 is an interior point (has an open ball in R2) in should be obvious, so it is open. But R2 also contains all of its limit points (why?), so it is closed.
 
Number Nine said:
But R2 also contains all of its limit points (why?), so it is closed.

Ok, well that's probably why she said it was beyond the scope of the PDE course. Guess I'll do some reading on limit points!

Thanks for your help!
 
romsofia said:
If we define a set, c, to be R^2, how is it open and closed?

The definitions I'm using:

Open set: Open set, O, is an open set if for all points x are in O, and we can find ONE B(x,ρ) such that B(x,ρ) is less than zero.
This looks like non-sense to me. What does it mean for a set to be "less than zero"? I think you mean "we can find at least one B(x,ρ) that is a subset of O". That's obviously true for R2 since, for any x in R2 B(x,ρ) is certainly a subset of R2.

Closed set: Compliment of an open set, AKA R^n/O.
R2 is the compliment of the empty set so it is sufficient to prove that the empty set is open. And that follows from the logical principal that "if P then Q" is true in the case that P is false, no matter whether Q is true of false. For the empty set, "if x is in O" is always false because the empty set contains NO points.

This isn't a HW question, I'm reviewing the analysis part of my PDE course from fall semester, and this is something extra she told us, and now I don't know how this could be true!

Thanks for the help.[/QUOTE]
 
HallsofIvy said:
This looks like non-sense to me. What does it mean for a set to be "less than zero"? I think you mean "we can find at least one B(x,ρ) that is a subset of O". That's obviously true for R2 since, for any x in R2 B(x,ρ) is certainly a subset of R2.

Hmm, this is out of my notes, so maybe I put zero, when I really meant O as I never understood why the ball at some element would have to be less than 0. My friend is borrowing my book right now, but I'd be able to check what the book says once I get it back.

x is an element, and ρ is the radius.

The book is Introductory to Patrial Differential Equations with Applications by Zachmanoglou and Thoe, if anyone has the book and can check.
 

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