A characterization of continuous images of compact ordered spaces and the Hahn-Mazurkiewicz problem

Bula, Witold D.; Turzański, Marian

doi:10.1016/0166-8641(86)90073-8

Cited by 6 publications

(5 citation statements)

References 6 publications

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“…By (1) of Theorem 1, X is an image of a first countable ordered compactum K. Thus, by Lemma 1, there are a first countable ordered compactum K and a reduced map f : K → X. By [3], there is a dense subset A of K such that for each x ∈ A, f −1 (f (x)) = {x}. A theorem of Qiao and Tall states that any first countable ordered space has a dense non-Archimedean subspace [30].…”

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confidence: 86%

Cardinal functions on continuous images of orderable compacta and applications

Farhat¹

2012

Preprint

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The class of Hausdorff spaces that are continuous images of compact orderable spaces is studied by analyzing the relationship between the elements of this class and compact orderable spaces in a back-and-forth fashion. Structure results for this class are then obtained, as well as continuum-theoretic embedding results. Applications to Boolean algebras are also demonstrated, specifically concerning the relationship between interval algebras and pseudotree algebras.

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confidence: 86%

Cardinal functions on continuous images of orderable compacta and applications

Farhat¹

2012

Preprint

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“…Since f is irreducible, K contains no locally countable points; in particular, K has no isolated points. We first extend a result of [2] to the effect that under these conditions, f is 1-to-1 on a dense subset of K.…”

Section: Theorem 1 Assume Sh Each Uncountable Monotonically Normal Co...mentioning

confidence: 98%

The basis problem for subspaces of monotonically normal compacta

Farhat

2015

Topology and its Applications

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“…The first characterizations of continuous images of non-metric arcs were given by Bula and Turzański [1] and by Nikiel [8]. For a survey of the problem, the reader is referred to the excellent survey paper of Treybig and Ward [13].…”

Section: Introductionmentioning

confidence: 99%

“…The classical Hahn-Mazurkiewicz Theorem asserts that a metric continuum is the continuous image of the closed unit interval [0, 1] if and only if it is locally connected. In the non-metric case, the situation turns out to be quite complicated.…”

Section: Introductionmentioning

confidence: 99%

Lifting paths on quotient spaces

Daniel

Nikiel

Treybig

et al. 2009

Topology and its Applications

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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.

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A characterization of continuous images of compact ordered spaces and the Hahn-Mazurkiewicz problem

Cited by 6 publications

References 6 publications

Cardinal functions on continuous images of orderable compacta and applications

Cardinal functions on continuous images of orderable compacta and applications

The basis problem for subspaces of monotonically normal compacta

Lifting paths on quotient spaces

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