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 Post subject: Fixed point?Posted: Thu, 6 Oct 2011 00:47:59 UTC
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This one I don't have any clue how to do, a friend asked it:

If are topological spaces such that any self-maps have a fixed point, does this hold for ? If so why?

Edit: Oops, this should be the V wedge, the one where we have the coproduct and glue them at a single point.

Edit 2: bolded Edit 1

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 Post subject: Re: Fixed point?Posted: Thu, 6 Oct 2011 01:08:58 UTC
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Joined: Mon, 29 Dec 2008 17:49:32 UTC
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This one I don't have any clue how to do, a friend asked it:

If are topological spaces such that any self-maps have a fixed point, does this hold for ? If so why?

Edit: Oops, this should be the V wedge, the one where we have the coproduct and glue them at a single point.

Edit 2: bolded Edit 1

Lemma: If have the fixed point property (fpp), then the one-point union has fpp too.
Proof: Let with , and let p be the common point {x,y}. Let . Assuming , so WLOG . Now defined by , is a retract of to . So is continuous, hence fpp of X tells us f has a fixed point.

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 Post subject: Re: Fixed point?Posted: Thu, 6 Oct 2011 01:14:47 UTC
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Location: Austin, TX
outermeasure wrote:
This one I don't have any clue how to do, a friend asked it:

If are topological spaces such that any self-maps have a fixed point, does this hold for ? If so why?

Edit: Oops, this should be the V wedge, the one where we have the coproduct and glue them at a single point.

Edit 2: bolded Edit 1

Lemma: If have the fixed point property (fpp), then the one-point union has fpp too.
Proof: Let with , and let p be the common point {x,y}. Let . Assuming , so WLOG . Now defined by , is a retract of to . So is continuous, hence fpp of X tells us f has a fixed point.

But what if has range in ?

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 Post subject: Re: Fixed point?Posted: Thu, 6 Oct 2011 01:15:57 UTC
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Oh, nvm, that's why r is there, to fix that.

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