Some complexity results in topology and analysis

Tom 141 / 1992

Steve Jackson, R. Mauldin Fundamenta Mathematicae 141 (1992), 75-83 DOI: 10.4064/fm-141-1-75-83

Streszczenie

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.

Autorzy

  • Steve Jackson
  • R. Mauldin

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