Continuous images of arcs and inverse limit methods

Nikiel, J.; Tuncali, H. Murat; Tymchatyn, E. D.

doi:10.1090/memo/0498

Cited by 17 publications

(8 citation statements)

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“…The class of generalized arcs is precisely the class of linearly orderable continua, each generalized arc admitting exactly two compatible linear orders. The class of (continuous images of) generalized arcs has been extensively studied over the years (see [10], [11], [14]), the most well-known results in this area being that any two arcs are homeomorphic (to the standard closed unit interval on the real line), and (Hahn-Mazurkiewicz) that a Hausdorff space is a continuous image of an arc if and only if that space is a locally connected metrizable continuum. In this paper, a continuation of [3], we study the model-theoretic topology of generalized arcs, in particular, the "dualized model theory" of these spaces.…”

Section: Introduction and Outline Of Resultsmentioning

confidence: 99%

Co-elementary equivalence, co-elementary maps, and generalized arcs

Bankston

1997

Proc. Amer. Math. Soc.

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Abstract. By a generalized arc we mean a continuum with exactly two non-separating points; an arc is a metrizable generalized arc. It is well known that any two arcs are homeomorphic (to the real closed unit interval); we show that any two generalized arcs are co-elementarily equivalent, and that co-elementary images of generalized arcs are generalized arcs. We also show that if f : X → Y is a function between compacta and if X is an arc, then f is a co-elementary map if and only if Y is an arc and f is a monotone continuous surjection.

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Section: Introduction and Outline Of Resultsmentioning

confidence: 99%

Co-elementary equivalence, co-elementary maps, and generalized arcs

Bankston

1997

Proc. Amer. Math. Soc.

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“…Proof. By Mardeśič [15] and by Nikiel, Tuncali, and Tymchatyn (Theorem 9.4 of [19]), there is a σ-directed inverse system Y = {Y a , q ab , A} such that each Y a is compact metric, each q ab is onto, and X = lim ← − Y. For each a ∈ A, let q a : X → Y a denote the natural projection.…”

Section: Inverse Limits Of Separable Continuamentioning

confidence: 99%

Hereditary separability in Hausdorff continua

Daniel

Tuncali

2013

Appl.Gen.Topol.

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We consider those Hausdorff continua S such that each separable subspace of S is hereditarily separable. Due to results of Ostaszewski and Rudin, respectively, all monotonically normal spaces and therefore all continuous Hausdorff images of ordered compacta also have this property. Our study focuses on the structure of such spaces that also possess one of various rim properties, with emphasis given to rim-separability. In so doing we obtain analogues of results of M. Tuncali and I. Lončar, respectively.

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“…In connection with [16,Theorem 5.1] it is naturally to ask the following question. Other basic notions, including approximate mapping and the limit of an approximate inverse system are defined as in [11,13].…”

Section: Proof (A)⇒(b) Obvious Since There Exists the Natural Projementioning

confidence: 99%