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Abstract. We prove that for each n ≥ 1 the set of all surjective continuumwise injective maps from an n-dimensional continuum onto an LC n−1 -continuum with the disjoint (n−1, n)-cells property is a dense G δ -subset of the space of all surjective maps. This generalizes a result of Espinoza and the second author [5].

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A surjective continuous map f : [0, 1] → X is called an arcwise increasing map if for any two closed subintervalsA continuum X is said to admit an arcwise increasing map if there is an arcwise increasing map onto X. It is shown that any Peano continuum with no free arcs admits an arcwise increasing map, and a characterization of graphs and dendrites that admit arcwise increasing maps is given.