A mapping characterization of Peano spaces

Harrold, O. G.

doi:10.1090/s0002-9904-1942-07729-9

Cited by 3 publications

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“…Originating with Hilbert's observation (1891) [33] that the square is a continuous image of the circle so that each point is visited at most three times, the natural question arises which properties beyond 'Peano' are needed to guarantee the existence of well-behaved such continuous surjections. Achieving additional control over the surjections from the circle, however, is a notorious open problem in continuum theory discussed, for example, in Nöbling (1933) [46], Harrold (1940Harrold ( , 1942 [31,32], Ward (1977) [56], Treybig & Ward (1981) [54, §4], Treybig (1983) [53], and Bula, Nikiel & Tymchatyn (1994) [12]. The latter six authors were particularly interested in the existence of strongly irreducible maps from the circle, continuous surjections g : S 1 → X such that for any proper closed subset A S 1 we have g(A) g(S 1 ).…”

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confidence: 99%

Eulerian Spaces

Gartside¹,

Pitz²

2019

Preprint

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We develop a unified theory of Eulerian spaces by combining the combinatorial theory of infinite, locally finite Eulerian graphs as introduced by Diestel and Kühn with the topological theory of Eulerian continua defined as irreducible images of the circle, as proposed by Bula, Nikiel and Tymchatyn.First, we clarify the notion of an Eulerian space and establish that all competing definitions in the literature are in fact equivalent. Next, responding to an unsolved problem of Treybig and Ward from 1981, we formulate a combinatorial conjecture for characterising the Eulerian spaces, in a manner that naturally extends the characterisation for finite Eulerian graphs. Finally, we present far-reaching results in support of our conjecture which together subsume and extend all known results about the Eulerianity of infinite graphs and continua to date. In particular, we characterise all one-dimensional Eulerian spaces.

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Eulerian Spaces

Gartside¹,

Pitz²

2019

Preprint

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Eulerian Spaces

Gartside,

Pitz

2023

Memoirs of the AMS

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We develop a unified theory of Eulerian spaces by combining the combinatorial theory of infinite, locally finite Eulerian graphs as introduced by Diestel and Kühn with the topological theory of Eulerian continua defined as irreducible images of the circle, as proposed by Bula, Nikiel and Tymchatyn. First, we clarify the notion of an Eulerian space and establish that all competing definitions in the literature are in fact equivalent. Next, responding to an unsolved problem of Treybig and Ward from 1981, we formulate a combinatorial conjecture for characterising the Eulerian spaces, in a manner that naturally extends the characterisation for finite Eulerian graphs. Finally, we present far-reaching results in support of our conjecture which together subsume and extend all known results about the Eulerianity of infinite graphs and continua to date. In particular, we characterise all one-dimensional Eulerian spaces.

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The Königsberg Bridge Problem for Peano Continua

Bula¹,

Nikiel²,

Tymchatyn³

1994

Can. j. math.

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Peano continua which are images of the unit interval [0,1] or the circle S under a continuous and irreducible map are investigated. Necessary conditions for a space to be the irreducible image of [0,1] are given, and it is conjectured that these conditions are sufficient as well. Also, various results on irreducible images of [0,1] and S are given within some classes of regular curves. Some of them involve inverse limits of inverse sequences of Euler graphs with monotone bonding maps.

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A mapping characterization of Peano spaces

Cited by 3 publications

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Eulerian Spaces

Eulerian Spaces

Eulerian Spaces

The Königsberg Bridge Problem for Peano Continua

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