Search results for "Images of arcs"
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Continuous images of arcs: Extensions of Cornette's Theorem
2015
In [J.L. Cornette “Image of a Hausdorff arc” is cyclically extensible and reducible Trans. Am. Math. Soc., 199 (1974), pp. 253–267], Cornette proved that a locally connected Hausdorff continuum X is the continuous image of an arc if and only if each of its cyclic elements is the continuous image of an arc. Cyclic elements form a closed null cover of X by retracts of X. We generalize Cornette's result to closed null covers of X with a dendritic structure. We give examples to show that some of our conditions are necessary and we pose some open questions.
Lifting paths on quotient spaces
2009
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) Hahn–Mazurkiewicz 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 firs…