peyman Euclid
Posts : 49 Join date : 20091106 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. 

Bruno Admin
Posts : 184 Join date : 20090915 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. 

peyman Euclid
Posts : 49 Join date : 20091106 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! 

Mohammad Descartes
Posts : 100 Join date : 20091105 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 

Bruno Admin
Posts : 184 Join date : 20090915 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). 

nick Euclid
Posts : 95 Join date : 20090915 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 ? 
