Components and inductive dimensions of compact spaces

Krzempek, Jerzy

doi:10.1007/s10587-010-0027-1

Cited by 6 publications

(5 citation statements)

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“…We shall recall the definition of the space Z(X, Y ), and investigate its properties (cf. [10]). To begin, write S X for the family of all subsets of X that are either finite (so ∅ ∈ S X ), or homeomorphic to A ℵ 0 .…”

Section: Spreading Out Compact Spaces In a Plankmentioning

confidence: 99%

“…It follows from Lemma 2.2 that Remark 4.7. The construction in the above proof is essentially the same as in the proof of Theorem 5 in [10] (see Remarks 3-4 therein), which yields a compact Fréchet space X C,α with dim X C,α = n, trind X C,α = trInd X C,α = α, and with components homeomorphic to C. The proofs of Lemmas 4.1 and 4.3 in the present paper are more complex than the proofs of corresponding Lemmas 6 and 7 in [10].…”

Section: Spreading Out Compact Spaces In a Plankmentioning

confidence: 99%

“…We may add at this place that Lemma 7 in [10] needs one more assumption (necessary but missed out): the space B in that statement should be a non-degenerate continuum (then each component of a non-empty open subspace is uncountable). Proposition 2.1 and Theorem 4.6 yield Corollary 4.8.…”

Section: Spreading Out Compact Spaces In a Plankmentioning

confidence: 99%

“…Let n be a natural number, α an ordinal, and 1 ≤ n ≤ α. In [10] we have combined constructions by P. Vopěnka [12] and V. A. Chatyrko [3], and have assigned a compact space Z(X, Y ) to any pair of non-empty compact spaces X, Y . Each component of Z(X, Y ) is homeomorphic to a component of X or Y .…”

Section: Introductionmentioning

confidence: 99%

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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: Spreading Out Compact Spaces In a Plankmentioning

confidence: 99%

Section: Spreading Out Compact Spaces In a Plankmentioning

confidence: 99%

Section: Spreading Out Compact Spaces In a Plankmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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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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“…For some other maps f : X → Y , even Ind X −Ind f X −Ind f > 1. For every pair of natural numbers m > n ≥ 1, the present author [33] has constructed a compact space X m,n such that Ind X m,n = m and every component of X m,n is homeomorphic to the n-dimensional cube [0, 1] n . In consequence, if D stands for the decomposition of X m,n into its components, and f K : X m,n → X m,n /D is the natural quotient map (it is not fully closed), then Ind X m,n − Ind X m,n /D − Ind f K = m − n.…”

Section: −1mentioning

confidence: 99%

Fully closed maps and non-metrizable higher-dimensional Anderson–Choquet continua

Krzempek

2010

Colloq. Math.

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Abstract. Fedorchuk's fully closed (continuous) maps and resolutions are applied in constructions of non-metrizable higher-dimensional analogues of Anderson, Choquet, and Cook's rigid continua. Certain theorems on dimensionlowering maps are proved for inductive dimensions and fully closed maps from spaces that need not be hereditarily normal, and some examples of continua have non-coinciding dimensions.Fully closed (continuous) maps and resolutions appear in numerous constructions (see S. Watson [48], V.V. Fedorchuk [26] for surveys), in particular, in constructions of homogeneous spaces with non-coinciding dimensions. In this paper we apply the maps in order to obtain examples of continua with strong, hereditary rigidity properties.A non-degenerate continuum X is called -an Anderson-Choquet continuum if every non-degenerate subcontinuum P of X has exactly one embedding P → X, the identity id P ; -a Cook continuum if every non-degenerate subcontinuum P of X has exactly one non-constant map P → X, the identity id P .Examples were constructed-respectively-by R. [32] for more references). In [32] the present author constructed a metric, n-dimensional (arbitrary n > 1), hereditarily indecomposable continuum no two of whose disjoint n-dimensional subcontinua are homeomorphic, but he was not able to ensure that the continuum be Anderson-Choquet.2000 Mathematics Subject Classification. 54F15, 54F45.

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The Gap Between the Covering and the Inductive Dimensions of Compact Hausdorff Spaces

Charalambous

2019

Dimension Theory

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Components and inductive dimensions of compact spaces

Cited by 6 publications

References 6 publications

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

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

Fully closed maps and non-metrizable higher-dimensional Anderson–Choquet continua

The Gap Between the Covering and the Inductive Dimensions of Compact Hausdorff Spaces

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