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 Fixed point

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peyman
Euclid
Euclid
peyman

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Join date : 2009-11-06
Age : 95
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Fixed point Empty
PostSubject: Fixed point   Fixed point EmptyWed 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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Bruno

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Join date : 2009-09-15
Age : 32
Location : the infinite, frictionless plane of uniform density

Fixed point Empty
PostSubject: Re: Fixed point   Fixed point EmptyThu 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
Euclid
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peyman

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Age : 95
Location : dense in the universe

Fixed point Empty
PostSubject: Re: Fixed point   Fixed point EmptyThu 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
Descartes
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Mohammad

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Age : 35
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Fixed point Empty
PostSubject: Re: Fixed point   Fixed point EmptySat 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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Fixed point Empty
PostSubject: Re: Fixed point   Fixed point EmptySat 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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nick

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Age : 57
Location : Alexandria

Fixed point Empty
PostSubject: Re: Fixed point   Fixed point EmptyMon 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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