Lifting paths on quotient spaces

Abstract

Let X be a compactum and G an upper semi-continuous decomposition of X such that each element of G is the continuous image of an ordered compactum. If the quotient space X/G is the continuous image of an ordered compactum, under what conditions is X also the continuous image of an ordered compactum? Examples around the (non-metric) HahnMazurkiewicz Theorem show that one must place severe conditions on G if one wishes to obtain positive results. We prove that the compactum X is the image of an ordered compactum when each g ∈ G has 0-dimensional boundary. We also consider the case when G has only countably many non-degenerate elements. These results extend earlier work of the first named author in a number of ways.

Decompositions of cyclic elements of locally connected continua

“…Theorem (Theorem 1 of Daniel, Nikiel, Treybig, Tuncali, and Tymchatyn [4]). Let Z be a locally connected continuum and H an upper semicontinuous decomposition of Z such that (i) each g ∈ H is connected and has zero-dimensional boundary, (ii) each g ∈ H is a continuous image of an ordered compactum.…”

“…We chose to include the proof above because it reveals slightly more of the structure of the elements of G, but an alternate proof is given simply by recalling the result of Cornette [1] and then directly applying the result above from [4].…”