# Some complexity results in topology and analysis

Fundamenta Mathematicae (1992)

• Volume: 141, Issue: 1, page 75-83
• ISSN: 0016-2736

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## Abstract

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If X is a compact metric space of dimension n, then K(X), the n- dimensional kernel of X, is the union of all n-dimensional Cantor manifolds in X. Aleksandrov raised the problem of what the descriptive complexity of K(X) could be. A straightforward analysis shows that if X is an n-dimensional complete separable metric space, then K(X) is a ${\Sigma }_{2}^{1}$ or PCA set. We show (a) there is an n-dimensional continuum X in ${ℝ}^{n}+1$ for which K(X) is a complete ${\Pi }_{1}^{1}$ set. In particular, $K\left(X\right)\in {\Pi }_{1}^{1}-{\Sigma }_{1}^{1}$; K(X) is coanalytic but is not an analytic set and (b) there is an n-dimensional continuum X in ${ℝ}^{n}+2$ for which K(X) is a complete ${\Sigma }_{2}^{1}$ set. In particular, $K\left(X\right)\in {\Sigma }_{2}^{1}-{\Pi }_{2}^{1}$; K(X) is PCA, but not CPCA. It is also shown the Lebesgue measure as a function on the closed subsets of [0,1] is an explicit example of an upper semicontinuous function which is not countably continuous.

## How to cite

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Jackson, Steve, and Mauldin, R.. "Some complexity results in topology and analysis." Fundamenta Mathematicae 141.1 (1992): 75-83. <http://eudml.org/doc/211952>.

@article{Jackson1992,
abstract = {If X is a compact metric space of dimension n, then K(X), the n- dimensional kernel of X, is the union of all n-dimensional Cantor manifolds in X. Aleksandrov raised the problem of what the descriptive complexity of K(X) could be. A straightforward analysis shows that if X is an n-dimensional complete separable metric space, then K(X) is a $Σ_2^1$ or PCA set. We show (a) there is an n-dimensional continuum X in $ℝ^n+1$ for which K(X) is a complete $Π_1^1$ set. In particular, $K(X) ∈ Π_1^1-Σ_1^1$; K(X) is coanalytic but is not an analytic set and (b) there is an n-dimensional continuum X in $ℝ^n+2$ for which K(X) is a complete $Σ_2^1$ set. In particular, $K(X) ∈ Σ_2^1-Π_2^1$; K(X) is PCA, but not CPCA. It is also shown the Lebesgue measure as a function on the closed subsets of [0,1] is an explicit example of an upper semicontinuous function which is not countably continuous.},
author = {Jackson, Steve, Mauldin, R.},
journal = {Fundamenta Mathematicae},
keywords = {cantor manifold; dimensional kernel; projective set; countably continuous; upper semicontinuous; -dimensional Cantor manifold; descriptive complexity},
language = {eng},
number = {1},
pages = {75-83},
title = {Some complexity results in topology and analysis},
url = {http://eudml.org/doc/211952},
volume = {141},
year = {1992},
}

TY - JOUR
AU - Jackson, Steve
AU - Mauldin, R.
TI - Some complexity results in topology and analysis
JO - Fundamenta Mathematicae
PY - 1992
VL - 141
IS - 1
SP - 75
EP - 83
AB - If X is a compact metric space of dimension n, then K(X), the n- dimensional kernel of X, is the union of all n-dimensional Cantor manifolds in X. Aleksandrov raised the problem of what the descriptive complexity of K(X) could be. A straightforward analysis shows that if X is an n-dimensional complete separable metric space, then K(X) is a $Σ_2^1$ or PCA set. We show (a) there is an n-dimensional continuum X in $ℝ^n+1$ for which K(X) is a complete $Π_1^1$ set. In particular, $K(X) ∈ Π_1^1-Σ_1^1$; K(X) is coanalytic but is not an analytic set and (b) there is an n-dimensional continuum X in $ℝ^n+2$ for which K(X) is a complete $Σ_2^1$ set. In particular, $K(X) ∈ Σ_2^1-Π_2^1$; K(X) is PCA, but not CPCA. It is also shown the Lebesgue measure as a function on the closed subsets of [0,1] is an explicit example of an upper semicontinuous function which is not countably continuous.
LA - eng
KW - cantor manifold; dimensional kernel; projective set; countably continuous; upper semicontinuous; -dimensional Cantor manifold; descriptive complexity
UR - http://eudml.org/doc/211952
ER -

## References

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1. [A] P. S. Aleksandrov, On the dimension of closed sets, Uspekhi Mat. Nauk 4 (6) (1949), 17-88 (in Russian).
2. S. I. Adyan and P. S. Novikov, On a semicontinuous function, Moskov. Gos. Ped. Inst. Uchen. Zap. 138 (3) (1958), 3-10 (in Russian). Zbl1296.26015
3. [B] B. L. Brechner, On the dimensions of certain spaces of homeomorphisms, Trans. Amer. Math. Soc. 121 (1966), 516-548. Zbl0151.30601
4. [E] R. Engelking, Dimension Theory, PWN and North-Holland, Warszawa-Amsterdam 1978.
5. [M] J. van Mill, n-dimensional totally disconnected topological groups, Math. Japon. 32 (1987), 267-273. Zbl0622.22003
6. [P] R. Pol, An n-dimensional compactum which remains n-dimensional after removing all Cantor n-manifolds, Fund. Math. 136 (1990), 127-131. Zbl0713.54039

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