Some complexity results in topology and analysis

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 -