Some complexity results in topology and analysis

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 ${\displaystyle {\Sigma }_2^1}$ or PCA set. We show (a) there is an n-dimensional continuum X in ${\displaystyle {ℝ}^n+1}$ for which K(X) is a complete ${\displaystyle {\Pi }_1^1}$ set. In particular, ${\displaystyle K\left(X\right)\in {\Pi }_1^1-{\Sigma }_1^1}$; K(X) is coanalytic but is not an analytic set and (b) there is an n-dimensional continuum X in ${\displaystyle {ℝ}^n+2}$ for which K(X) is a complete ${\displaystyle {\Sigma }_2^1}$ set. In particular, ${\displaystyle K\left(X\right)\in {\Sigma }_2^1-{\Pi }_2^1}$; 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.