Dendrites and light mappings

Abstract

It is shown that a metric continuum X X is a dendrite if and only if for every compact space (continuum) Y Y and for every light confluent mapping f : Y → f ( Y ) f: Y \to f(Y) such that X ⊂ f ( Y ) X \subset f(Y) there is a copy X ′ X’ of X X in Y Y for which the restriction f | X ′ : X ′ → X f|X’: X’ \to X is a homeomorphism. As a corollary it follows that only dendrites have the lifting property with respect to light confluent mappings. Other classes of mappings f f are also discussed. This is a continuation of a previous study by the authors (2000), where open mappings f f were considered.