Rigid continua and transfinite inductive dimension

Charalambous, Michael G.; Krzempek, Jerzy

doi:10.1016/j.topol.2010.03.008

Cited by 8 publications

(8 citation statements)

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“…As his resolution map B 1 → P is atomic and fully closed, and every point-inverse of the map is homeomorphic to P, B 1 is certainly chainable. Note that in [6] we have shown that every resolution S P (P) in the stronger sense has ind S P (P) = Ind 0 S P (P) = 2. It will follow from Corollary 5.3 in the present paper that Dg S P (P) = 2, too.…”

Section: Lemmamentioning

confidence: 76%

“…Resolutions were introduced by Fedorchuk in [15], and a thorough treatment of fully closed maps and resolutions can be found in [18]. A variant of resolutions for which it is relatively easy to compute inductive dimensions was recently introduced in [6], where all results quoted below can be found.…”

Section: Proof Consider Disjoint Closed Subsetsmentioning

confidence: 99%

“…Proof. In view of Corollary 2.6, f is ring-like, and X is chainable by [6,Proposition 3]). The assertion on end points is clear from the proof of that result.…”

Section: Inverse Invariance Of Chainability a Chain Is A Finite Collmentioning

confidence: 99%

“…(3) Theorem 5.7 provides a positive answer in the realm of first countable spaces to Pasynkov's question [27] whether for every ordinal α, there exists a chainable continuum with trind = α. Note that in [6] we have constructed a first countable, separable, HD chainable continuum S α with trind S α = trInd 0 S α = α, for each α of cardinality ≤ c.…”

Section: Remarks (1)mentioning

confidence: 99%

“…If in the above definition of Ind, we stipulate that the set A is a singleton, we obtain the definition of ind. In [5,6,22] we used Ind 0 as a tool for estimating ind and Ind. Dg or Dimensionsgrad is Brouwer's original definition of dimension [3].…”

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confidence: 99%

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On Dimensionsgrad, resolutions, and chainable continua

Charalambous

Krzempek

2010

Fund. Math.

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There exist very few examples of spaces exhibiting pathologies of Brouwer's Dimensionsgrad, Dg. For each natural number n ≥ 1 and each pair of ordinals α, β with n ≤ α ≤ β ≤ ω(c +), where ω(c +) is the first ordinal of cardinality c + , we construct a continuum S n,α,β such that (a) dim S n,α,β = n; (b) trDg S n,α,β = trDg 0 S n,α,β = α; (c) trind S n,α,β = trInd0 S n,α,β = β; (d) if β < ω(c +), then S n,α,β is separable and first countable; (e) if n = 1, then S n,α,β can be made chainable or hereditarily decomposable; (f) if α = β < ω(c +), then S n,α,β can be made hereditarily indecomposable; (g) if n = 1 and α = β < ω(c +), then S n,α,β can be made chainable and hereditarily indecomposable. In particular, we answer the question raised by Chatyrko and Fedorchuk whether every non-degenerate chainable space has Dimensionsgrad equal to 1. Moreover, we establish results that enable us to compute the Dimensionsgrad of a number of spaces constructed by Charalambous, Chatyrko, and Fedorchuk.

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Section: Lemmamentioning

confidence: 76%

Section: Proof Consider Disjoint Closed Subsetsmentioning

confidence: 99%

“…Proof. In view of Corollary 2.6, f is ring-like, and X is chainable by [6,Proposition 3]). The assertion on end points is clear from the proof of that result.…”

Section: Inverse Invariance Of Chainability a Chain Is A Finite Collmentioning

confidence: 99%

Section: Remarks (1)mentioning

confidence: 99%

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confidence: 99%

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