Boundary of an open set in R2 is a limit point?

In summary, it is not true that any points on the boundary of an open set are limit points of the set.
  • #1
dumbQuestion
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I have kind of a simple point set topology question. If I am in ℝ2 and I have a connected open set, call it O, then is it true that all points on the boundary ∂O are limit points of O? I guess I'm stuck envisioning as O as, at least homeomorphic, to an open disk of radius epsilon. So it seems obvious that any points on the boundary would be limit points. But is that true in general?
 
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  • #2
Let [itex]U\subseteq \mathbb{R}^{n}[/itex] be connected and open and non empty. [itex]p\in \partial U[/itex] if and only if every neighborhood of [itex]p[/itex] contains both a point in [itex]U[/itex] (and in [itex]\mathbb{R}^{n}\setminus U[/itex] but we don't care about that here). Let [itex]p\in \partial U[/itex] and assume there exists a neighborhood [itex]V[/itex] of [itex]p[/itex] in [itex]\mathbb{R}^{n}[/itex] such that [itex]V\cap U = \left \{ p \right \}[/itex] (we know of course that [itex]U\supset \left \{ p \right \}[/itex]). This implies [itex]\left \{ p \right \}[/itex] is a non - empty proper clopen subset of [itex]U[/itex] which is a contradiction because [itex]U[/itex] is connected. Thus, [itex]p[/itex] is a limit point of [itex]U[/itex].
 
  • #3
thank you very much!
 
  • #4
dumbQuestion said:
thank you very much!
Should work for any Hausdorff connected space and not just euclidean space as far as I can see. Was there a particular reason for this question or did it just pop into your head for fun or something= D? Cheers!
 
  • #5
WannabeNewton said:
Let [itex]U\subseteq \mathbb{R}^{n}[/itex] be connected and open and non empty. [itex]p\in \partial U[/itex] if and only if every neighborhood of [itex]p[/itex] contains both a point in [itex]U[/itex] (and in [itex]\mathbb{R}^{n}\setminus U[/itex] but we don't care about that here). Let [itex]p\in \partial U[/itex] and assume there exists a neighborhood [itex]V[/itex] of [itex]p[/itex] in [itex]\mathbb{R}^{n}[/itex] such that [itex]V\cap U = \left \{ p \right \}[/itex] (we know of course that [itex]U\supset \left \{ p \right \}[/itex]). This implies [itex]\left \{ p \right \}[/itex] is a non - empty proper clopen subset of [itex]U[/itex] which is a contradiction because [itex]U[/itex] is connected. Thus, [itex]p[/itex] is a limit point of [itex]U[/itex].

Unfortunately this does not work since open sets will not contain their boundary points. Luckily that observation gives us a way to fix the proof. All we now have to note is that the boundary point condition implies that every nbhd of p non-trivially intersects U - {p} = U.
 
  • #6
jgens said:
Unfortunately this does not work since open sets will not contain their boundary points.
Totally missed that detail haha. Maybe the OP didn't mean to say open but another class of subsets because he was trying to use connectedness explicitly. Anyways, I got to go to class now so cheers!
 

1. What is the definition of an open set in R2?

An open set in R2 is a set of points that are all contained within a given region. This region is open if any point within it can be surrounded by a circle of any size, and all points within this circle are also contained within the region.

2. How is the boundary of an open set in R2 defined?

The boundary of an open set in R2 is the set of points that lie on the edge of the given region. These points are neither completely within the region nor completely outside of it.

3. Can the boundary of an open set in R2 contain any limit points?

Yes, the boundary of an open set in R2 can contain limit points. This is because the boundary itself is defined as the set of points that are neither completely within the region nor completely outside of it. Therefore, points that are close to the boundary but not contained within it can be considered limit points.

4. How is the boundary of an open set in R2 different from the boundary of a closed set?

The boundary of an open set in R2 only contains points that are on the edge of the given region, while the boundary of a closed set can contain points both on the edge and within the set. In other words, the boundary of an open set is the boundary of its complement, while the boundary of a closed set is the boundary of the set itself.

5. Why is understanding the boundary of an open set in R2 important in mathematics?

Understanding the boundary of an open set in R2 is important in mathematics because it helps us to define and visualize different types of sets. It also allows us to distinguish between different types of points, such as interior, exterior, and boundary points, which can be useful in various mathematical proofs and applications.

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