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# Fixed point

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Euclid

Posts : 49
Join date : 2009-11-06
Age : 95
Location : dense in the universe

 Subject: 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.

Posts : 184
Join date : 2009-09-15
Age : 31
Location : the infinite, frictionless plane of uniform density

 Subject: 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. 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.

Euclid

Posts : 49
Join date : 2009-11-06
Age : 95
Location : dense in the universe

 Subject: 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!

Descartes

Posts : 100
Join date : 2009-11-05
Age : 35
Location : Right behind you

 Subject: 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

Posts : 184
Join date : 2009-09-15
Age : 31
Location : the infinite, frictionless plane of uniform density

 Subject: 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 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).

Euclid

Posts : 95
Join date : 2009-09-15
Age : 56
Location : Alexandria

 Subject: 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 ?

 Subject: Re: Fixed point

 Fixed point
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