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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 consider a set, L, of lines in ${ℝ}^n$ and a partition of L into some number of sets: $L=L_1\cup ...\cup L_p$. We seek a corresponding partition ${ℝ}^n=S_1\cup ...\cup S_p$ such that each line l in $L_i$ meets the set $S_i$ in a set whose cardinality has some fixed bound, ${\omega }_{\tau }$. We determine equivalences between the bounds on the size of the continuum, $2^{\omega }\le {\omega }_{\theta }$, and some relationships between p, ${\omega }_{\tau }$ and ${\omega }_{\theta }$.

Given a partition P:L → ω of the lines in ${ℝ}^n$, n ≥ 2, into countably many pieces, we ask if it is possible to find a partition of the points, $Q:{ℝ}^n\to \omega$, so that each line meets at most m points of its color. Assuming Martin’s Axiom, we show this is the case for m ≥ 3. We reduce the problem for m = 2 to a purely finitary geometry problem. Although we have established a very similar, but somewhat simpler, version of the geometry conjecture, we leave the general problem open. We consider also various generalizations...

We consider the Borel structures on ordinals generated by their order topologies and provide a complete classification of all ordinals up to Borel isomorphism in ZFC. We also consider the same classification problem in the context of AD and give a partial answer for ordinals ≤ω₂.