Local selections and local dendrites

Abstract

Let X be a Peano continuum, C( X) its space of subcontinua, and C( X, ε) the space of subcontinua of diameter less than ε. A selection on some subspace of C( X) is a continuous choice function; the selection σ is rigid if σ( A) ϵ B ⊂ A implies σ( A) = σ( B). It is shown that X is a local dendrite (contains at most one simple closed curve) if and only if there exists ε > 0 such that C( X, ε) admits a selection (rigid selection). Further, C( X) admits a local selection at the subcontinuum A if and only if A has a neighborhood (relative to the space C( X)) which contains no cyclic local dendrite; moreover, that local selection may be chosen to be a constant.