Shadow 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 1Lemma: 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.