Fixed point set characterizations of Peano continua and absolute retracts (Q1873297)

If \(Z\) is a compactum (that is, a compact metric space), then \(C(Z)\) denotes the space of all subcontinua of \(Z\) with the Hausdorff metric and for \(\delta\geq 0\), \(C_\delta(Z)\) its subspace of all continua whose diameter is at most \(\delta\). A map \(Z\mapsto C(Z)\) is called a \textsl{continuum-valued map}. A point \(p\in Z\) is said to be a \textsl{fixed point of the continuum-valued map F} if \(p\in F(p)\). A set \(A\subseteq Z\) is then said to be a \textsl{continuum-valued fixed point set of Z} if it is the fixed point set of some continuum-valued function \(F:Z\to C(Z)\). A compactum \(A\) is said to be an \textsl{absolute fixed point set} provided that whenever \(A\) is embedded as a subspace \(A^\prime\) of a compactum \(Z\), then \(A^\prime\) is the fixed point set of a map \(g:Z\to Z\). The family of all (necessarily compact) absolute fixed point sets is denoted by \textsl{AFS}. Finally, a compactum \(A\) is an \(\varepsilon\)\textsl{-MAFS} (respectively a \textsl{0-MAFS}) provided that whenever \(A\) is embedded as a subspace \(A^\prime\) of a compactum \(Z\), then for every \(\delta>0\) (respectively \(\delta=0\)) there is some map \(F_\delta:Z\to C_\delta(Z)\) whose fixed point set is \(A^\prime\). This paper, which extends results of \textit{J. R. Martin} [Fundam. Math. 112, 159--164 (1981; Zbl 0448.54009)], deals with the relationships between Peano continua, absolute retracts, absolute fixed point sets (which are precisely the \textsl{0-MAFS}), \(\varepsilon\)\textsl{-MAFS} and two other more general families of spaces (\textsl{weak 0-MAFS} and \textsl{weak \(\varepsilon\)-MAFS}). One of the two main theorems states: If \(A\) is either a one-dimensional continuum or a planar continuum then the following are equivalent: 1) \(A\) is an absolute retract, 2) \(A\) is an \(AFS\)-space, 3) \(A\) is an \(\varepsilon\)\textsl{-MAFS}-space.