Subject: Re: question about fixed points
Date: Mon, 19 Nov 2001 18:10:37 +0100 (CET)
From: Marek Golasinski <marek@mat.uni.torun.pl>
To: Don Davis <dmd1@lehigh.edu>

You wrote:
===========
Suppose that X is a nonempty, compact, contractible, metrizable subset
of a locally convex linear topological space.
(i) Is X an absolute neighborhood retract? (If so, is there a reference
for such a result?)

(ii) If the answer to (i) is "not necessarily," then suppose in addition
that F is a point to set mapping on X with a closed graph such that F(x)
is nonempty and contractible for every x in X. Must F possess a fixed
point?
(Obviously an affirmative answer to (i) implies an affirmative answer to
(ii).
===========

In the paper by Ronald J. Knill, "Cones, product and fixed points",
Fund.Math. LX (1967), 35-46
on the page 43 there is constructed a compact and contractible
subset $B\in R^3$ and Theorem 3.4 says:
Neither the cone $C(v;B)($ nor the product $B\times I$ have
the fixed point.
Therefore the answer for the question (ii) is negative even for
one-valeued mappings.

Marek Golasinski