If two maps on a space X,
which admits a fixed point index, have identical sets of fixed points and agree
on an open subset of X which contains the fixed point set, then the maps
have the same Lefschetz number. If the subset is closed, the con clusion is
no longer true in general. However, a theorem of Leray implies that some
kinds of maps on cartesian products of convexoid spaces which agree on
a certain closed subset of their common fixed point set do have the same
Lefschetz number, even though the maps may not be homotopic and may
not agree on any open set containing the fixed point set. The purpose of
this note is to prove a very general form of Leray‘s theorem for maps on
ANR’s.
