- Tytuł:
- Some complexity results in topology and analysis
- Autorzy:
-
Jackson, Steve
Mauldin, R. - Powiązania:
- https://bibliotekanauki.pl/articles/1215022.pdf
- Data publikacji:
- 1992
- Wydawca:
- Polska Akademia Nauk. Instytut Matematyczny PAN
- Tematy:
-
cantor manifold
dimensional kernel
projective set
countably continuous
upper semicontinuous - Opis:
- 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.
- Źródło:
-
Fundamenta Mathematicae; 1992, 141, 1; 75-83
0016-2736 - Pojawia się w:
- Fundamenta Mathematicae
- Dostawca treści:
- Biblioteka Nauki
Artykuł