Some properties of the induced map

Gentry, Karl R.

doi:10.4064/fm-66-1-55-59

Cited by 8 publications

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“…Then, we will illustrate how to adjust the construction to additionally fulfill property (2) of the statement. We point out that part of the argument below-deriving from exactness that the map h is open-can also be found in [7].…”

Section: The Combinatorics Of Universalitymentioning

confidence: 97%

“…Structural exactness is a natural generalization of the well studied notion of exactness. Recall that an amalgamation diagram, as in Definition 5.1, is exact if for every b ∈ B, c ∈ C with f (b) = g(c) there is d ∈ D so that f (d) = b and g (d) = c; see [7]. In the context of Proposition 5.1, structural exactness of C will allow us to strengthen the connectedness properties of the map h. Two-sided structural exactness together with the next property will additionally allow us to control isomorphism type of the fibers of h. Definition 5.2.…”

Section: Corollary 41 (Andersonmentioning

confidence: 99%

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A combinatorial model for the Menger curve

Panagiotopoulos

Solecki

2020

J. Topol. Anal.

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We represent the universal Menger curve as the topological realization |M| of the projective Fraïssé limit M of the class of all finite connected graphs. We show that M satisfies combinatorial analogues of the Mayer-Oversteegen-Tymchatyn homogeneity theorem and the Anderson-Wilson projective universality theorem. Our arguments involve only 0-dimensional topology and constructions on finite graphs. Using the topological realization M → |M|, we transfer some of these properties to the Menger curve: we prove the approximate projective homogeneity theorem, recover Anderson's finite homogeneity theorem, and prove a variant of Anderson-Wilson's theorem. The finite homogeneity theorem is the first instance of an "injective" homogeneity theorem being proved using the projective Fraïssé method. We indicate how our approach to the Menger curve may extend to higher dimensions.2010 Mathematics Subject Classification. 03C30, 54F15.

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Section: The Combinatorics Of Universalitymentioning

confidence: 97%

Section: Corollary 41 (Andersonmentioning

confidence: 99%

A combinatorial model for the Menger curve

Panagiotopoulos

Solecki

2020

J. Topol. Anal.

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“…Since each C i is nondegenerate, we may choose N sufficiently large so that the projection of each component onto Y N is nondegenerate. Since the projection mapping π N is monotone (see[8, Corollary to Theorem 10],[5, Theorem 5]), all pairwise intersections of the three subcontinua π N (C 1 ), π N (C 2 ), and π N (C 3 ) are precisely {π N (r)}. Thus, π N (r) is a ramification point of Y N .…”

mentioning

confidence: 91%

Monotone preimages of dendrites

Wright

2009

Topology and its Applications

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A counterexample to a theorem of J.J. Charatonik and K. Omiljanowski giving sufficient conditions for a dendrite to be contained in all of its monotone preimages is given, and a corrected version of the theorem is presented. An alternative proof is provided for a characterization of dendrites monotonely equivalent to a universal dendrite that was originally proved using the erroneous result. Finally, in response to a question by J.J. Charatonik, it is shown that a dendrite contained in all of its monotone preimages must have a discrete set of ramification points.

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“…For instance, while the inverse limit of open mappings is not necessarily open, the inverse limit (or even the weakly induced limit in the sense of Mioduszewski [15]) of confluent mappings is always confluent (cf. [9,17,10]). …”

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confidence: 99%

On confluently graph-like compacta

Oversteegen

Prajs

2003

Fund. Math.

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Abstract. For any class K of compacta and any compactum X we say that: (a) X is confluently K-representable if X is homeomorphic to the inverse limit of an inverse sequence of members of K with confluent bonding mappings, and (b) X is confluently K-like provided that X admits, for every ε > 0, a confluent ε-mapping onto a member of K. The symbol LC stands for the class of all locally connected compacta. It is proved in this paper that for each compactum X and each family K of graphs, X is confluently Krepresentable if and only if X is confluently K-like. We also show that for any compactum the properties of: (1) being confluently graph-representable, and (2) being 1-dimensional and confluently LC-like, are equivalent. Consequently, all locally connected curves are confluently graph-representable. We also conclude that all confluently arc-like continua are homeomorphic to inverse limits of arcs with open bonding mappings, and all confluently tree-like continua are absolute retracts for hereditarily unicoherent continua.

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Some properties of the induced map

Cited by 8 publications

References 0 publications

A combinatorial model for the Menger curve

A combinatorial model for the Menger curve

Monotone preimages of dendrites

On confluently graph-like compacta

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