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Some complexity results in topology and analysis

Steve JacksonR. Mauldin — 1992

Fundamenta Mathematicae

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 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...

On partitions of lines and space

Paul ErdösSteve JacksonR. Mauldin — 1994

Fundamenta Mathematicae

We consider a set, L, of lines in n and a partition of L into some number of sets: L = L 1 . . . L p . We seek a corresponding partition n = S 1 . . . S p such that each line l in L i meets the set S i in a set whose cardinality has some fixed bound, ω τ . We determine equivalences between the bounds on the size of the continuum, 2 ω ω θ , and some relationships between p, ω τ and ω θ .

On infinite partitions of lines and space

Paul ErdösSteve JacksonR. Mauldin — 1997

Fundamenta Mathematicae

Given a partition P:L → ω of the lines in n , n ≥ 2, into countably many pieces, we ask if it is possible to find a partition of the points, Q : n ω , 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...

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