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

We prove the following theorems: There exists an $\omega$-covering with the property $s_0$. Under $\mathrm{c}ov\left(𝒩\right)=$ there exists $X$ such that ${\forall }_{B\in ℬor}[B\cap X$ is not an $\omega$-covering or $X\setminus B$ is not an $\omega$-covering]. Also we characterize the property of being an $\omega$-covering.

We develop a theory of sharp measure zero sets that parallels Borel’s strong measure zero, and prove a theorem analogous to Galvin–Mycielski–Solovay theorem, namely that a set of reals has sharp measure zero if and only if it is meager-additive. Some consequences: A subset of $2^{\omega }$ is meager-additive if and only if it is $ℰ$-additive; if $f:2^{\omega }\to 2^{\omega }$ is continuous and $X$ is meager-additive, then so is $f\left(X\right)$.