Subject: Re: question about fixed points Date: Tue, 20 Nov 2001 08:29:04 +0100 (CET) From: Marek Golasinski To: Don Davis I enclose below a full answering to the question on the fixed points: ============== 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-valued mappings. Moreover, the subset $B\times I$ in $R^4$ is a counterexamaple to the question (i). Marek Golasinski