Some complexity results in topology and analysis

Jackson, Steve; Mauldin, R.

Fundamenta Mathematicae, Tome 141 (1992), p. 75-83 / Harvested from The Polish Digital Mathematics Library

Résumé

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, $K (X) \in Π_{1}^{1} - Σ_{1}^{1}$ ; 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, $K (X) \in Σ_{2}^{1} - Π_{2}^{1}$ ; 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.

Publié le : 1992-01-01

Zbl 0807.54013

EUDML-ID : urn:eudml:doc:211952

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     title = {Some complexity results in topology and analysis},
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Jackson, Steve; Mauldin, R. Some complexity results in topology and analysis. Fundamenta Mathematicae, Tome 141 (1992) pp. 75-83. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-fmv141i1p75bwm/

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