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Boundary of an open set in R2 is a limit point? 
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#1
Feb1913, 10:26 PM

P: 126

I have kind of a simple point set topology question. If I am in ℝ^{2} 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?



#2
Feb1913, 11:13 PM

C. Spirit
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P: 5,639

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
Feb2013, 12:14 AM

P: 126

thank you very much!



#4
Feb2013, 12:20 AM

C. Spirit
Sci Advisor
Thanks
P: 5,639

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



#5
Feb2013, 09:25 AM

P: 1,622




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