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Topic: Can one connected set pass through another? (Read 1080 times) 

ecoist
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Can one connected set pass through another?
« on: Apr 4^{th}, 2008, 9:24pm » 
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Let S be the square consisting of all points (x,y) in the plane such that 0<x,y<1. Can S be partitioned into two disjoint connected sets A and B such that A contains the vertical sides of S and B contains the horizontal sides of S?


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Obob
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Re: Can one connected set pass through another?
« Reply #1 on: Apr 5^{th}, 2008, 9:35am » 
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By connected, do you mean pathconnected?


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ecoist
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Re: Can one connected set pass through another?
« Reply #2 on: Apr 5^{th}, 2008, 10:15am » 
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No. The meaning here is: a set is connected if it is not the union of two disjoint, nonempty, open sets.


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Eigenray
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Re: Can one connected set pass through another?
« Reply #3 on: Apr 5^{th}, 2008, 5:20pm » 
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I would guess yes, since we should be able to use the axiom of choice to eliminate each possible pair of separating sets.


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ecoist
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Re: Can one connected set pass through another?
« Reply #4 on: Apr 5^{th}, 2008, 9:15pm » 
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Sorry, I am topologically retarded. I think the right definition of connected set is: a set is connected if it is a set which cannot be partitioned into two nonempty subsets such that each subset has no points in common with the set closure of the other. The set of two distinct points in the plane is obviously not a connected set, yet it cannot be partitioned into two disjoint nonempty open sets. A point x is in the set closure of a set S if every neighborhood of x meets S.


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Hooie
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Re: Can one connected set pass through another?
« Reply #5 on: Apr 22^{nd}, 2008, 6:02pm » 
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I believe you were right the first time. I think the two singletons would be open in the relative topology (a set U in a subspace A of X is open if it's of the form V intersect A, for some V open in X).


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ecoist
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Re: Can one connected set pass through another?
« Reply #6 on: Apr 22^{nd}, 2008, 8:49pm » 
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You are right, Hooie, but I wanted A and B to be connected in the plane, considered as a metric space. Hence, a set S in the plane is open if, for every point in S, some disk in the plane containing that point lies in S.


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Hooie
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Re: Can one connected set pass through another?
« Reply #7 on: Apr 23^{rd}, 2008, 3:16am » 
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Yes, I know, but what it means to be connected depends on whether you're talking about the whole space IxI or the subset {p} U {q}. A topological space X is connected if it can't be expressed as the disjoint union of two nonempty open subsets. A subset Y of a topological space X is connected if it's connected in the relative topology it inherits from X.


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Hippo
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Re: Can one connected set pass through another?
« Reply #8 on: Sep 26^{th}, 2008, 3:13pm » 
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What about set ysin (1/x) < 0.01 (unioned with x=0 and with x=1)?

« Last Edit: Sep 26^{th}, 2008, 3:15pm by Hippo » 
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