On inductive dimension modulo a simplicial complex

Charalambous, Michael G.; Krzempek, Jerzy

doi:10.1016/j.topol.2011.10.014

Cited by 2 publications

(4 citation statements)

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“…The dimension M -Ind 0 modulo a simplicial complex M [respectively: modulo an ANR M ] is defined similarly as M -Ind-in order that M -Ind 0 X ≤ α we stipulate that the M -partition P in the statement 1.3 (b) [respectively: 1.4(b')] is a zero set with M -Ind 0 P < α (see [2, p. 670]). It is easily shown by transfinite induction that M -Ind ≤ M -Ind 0 , and Theorem 1 in [2] may be summarised as follows: K-Ind 0 = |K|-Ind 0 for any simplicial complex K and all normal spaces. Thus, we have n ≤ |∂∆ k |-Ind Z i ≤ |∂∆ k |-Ind 0 Z i = ∂∆ k -Ind 0 Z i .…”

Section: Compact Spaces Withmentioning

confidence: 99%

“…5 Fedorchuk's original K-Ind X and L-Ind X in [8] are natural numbers, −1, or ∞. Following [2], we allow both K-Ind X and L-Ind X to be an infinite ordinal.…”

Section: Notation Basic Definitions and Factsmentioning

confidence: 99%

“…In the joint paper [2] with M. G. Charalambous we have constructed first countable and separable continua S n,α such that K-dim S n,α = n and K-Ind S n,α = α, where n ≥ 1 is any natural number, α ≥ n is any ordinal of cardinality at most c, and moreover n = 1 or the join |K| * |K| is non-contractible. This may be considered as a partial solution to the following.…”

Section: Introductionmentioning

confidence: 99%

“…We need the following claim: For each closed ∂∆ k -tuple F of X, there exists a ∂∆ k -partition P disjoint from X i . Indeed, Lemma 6 in [2] directly yields a map f : F → |∂∆ k | with F j ⊂ f −1 (K j ) for j = 0, . .…”

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On dimensions modulo a compact metric ANR and modulo a simplicial complex

Krzempek

2012

Topology and its Applications

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V. V. Fedorchuk has recently introduced dimension functions Kdim ≤ K-Ind and L-dim ≤ L-Ind, where K is a simplicial complex and L is a compact metric ANR. For each complex K with a non-contractible join |K| * |K| (we write |K| for the geometric realisation of K), he has constructed first countable, separable compact spaces with K-dim < K-Ind.In a recent paper we have combined an old construction by P. Vopěnka with a new construction by V. A. Chatyrko, and have assigned a certain compact space Z(X, Y ) to any pair of non-empty compact spaces X, Y . In this paper we investigate the behaviour of the four dimensions under the operation Z(X, Y ). This enables us to construct examples of compact Fréchet spaces which have K-dim < K-Ind, L-dim < L-Ind, or K-Ind < |K|-Ind, and (connected) components of which are metrisable. In particular, given a natural number n ≥ 1, an ordinal α ≥ n, and any metric continuum C with L-dim C = n, we obtain • a compact Fréchet space X C,α such that L-dim X C,α = n, L-Ind X C,α = α, and each component of X C,α is homeomorphic to C.If L * L is non-contractible, or n = 1 and L is non-contractible, then C can be a cube [0, 1] m for a certain natural number m = m(n, L).2000 Mathematics Subject Classification. Primary 54F45.

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Section: Compact Spaces Withmentioning

confidence: 99%

“…5 Fedorchuk's original K-Ind X and L-Ind X in [8] are natural numbers, −1, or ∞. Following [2], we allow both K-Ind X and L-Ind X to be an infinite ordinal.…”

Section: Notation Basic Definitions and Factsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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confidence: 99%

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