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 Fixed point

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peyman
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PostSubject: Fixed point   Wed Dec 02, 2009 8:50 pm

Let f:Y->Y be a continuous function on a triad Y. Does f have a fixed point? (Draw a big Y on paper. A triad is homeomorphic to what you drew!) Prove or disprove by elementary means.
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Bruno
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PostSubject: Re: Fixed point   Thu Dec 03, 2009 4:02 am

I can show easily that f has a fixed point if it is a homeomorphism! That's a big strengthening of the hypothesis, but it gives evidence that f has a fixed point even if it is just continuous. Smile

If f is a homeomorphism, then the "center" of the triad (the common point to the three segments) must be a fixed point, because there is no point on the triad having a neighbourhood homeomorphic to a neighbourhood of the center, other than the center itself.
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peyman
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PostSubject: Re: Fixed point   Thu Dec 03, 2009 4:02 pm

Bruno wrote:
..., because there is no point on the triad having a neighbourhood homeomorphic to a neighbourhood of the center, other than the center itself.

prove it!
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Mohammad
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PostSubject: Re: Fixed point   Sat Dec 19, 2009 4:32 pm

peyman wrote:
Bruno wrote:
..., because there is no point on the triad having a neighbourhood homeomorphic to a neighbourhood of the center, other than the center itself.

prove it!

Removing the center produces three pieces no other point does it Peyman! Bruno is right bounce
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Bruno
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PostSubject: Re: Fixed point   Sat Dec 19, 2009 4:45 pm

Mohammad wrote:
peyman wrote:
Bruno wrote:
..., because there is no point on the triad having a neighbourhood homeomorphic to a neighbourhood of the center, other than the center itself.

prove it!

Removing the center produces three pieces no other point does it Peyman! Bruno is right bounce

Peyman agrees with me, I spoke to him about this; but I think he was wondering why this property is a topological invariant (rather than doubting whether it is).
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nick
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PostSubject: Re: Fixed point   Mon Jan 25, 2010 7:41 am

Bruno wrote:
Peyman agrees with me, I spoke to him about this; but I think he was wondering why this property is a topological invariant (rather than doubting whether it is).
If is not invariant, we modify the topology and let the triple point as it is (invariant). So, what about the proof ?
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