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Topic: Can one connected set pass through another? (Read 1077 times) |
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ecoist
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Can one connected set pass through another?
« on: Apr 4th, 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 5th, 2008, 9:35am » |
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By connected, do you mean path-connected?
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ecoist
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Re: Can one connected set pass through another?
« Reply #2 on: Apr 5th, 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 5th, 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 5th, 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 22nd, 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 22nd, 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 23rd, 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 non-empty 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 26th, 2008, 3:13pm » |
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What about set |y-sin (1/x)| < 0.01 (unioned with x=0 and with x=1)?
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| « Last Edit: Sep 26th, 2008, 3:15pm by Hippo » |
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