Large entire cross-sections of second category sets in Rn+1
Burke, Maxim R.
Topology and its Applications
Journal Article
2007
154
1
215
240
10.1016/j.topol.2006.04.009
Mathematics and Statistics
By the Kuratowski-Ulam theorem, if A subset of Rn+1 = R-n x R is a Borel set which has second category intersection with every ball (i.e., is "everywhere second category"), then there is a y is an element of R such that the section A boolean AND (R-n x {y}) is everywhere second category in R-n x {y}. If A is not Borel, then there may not exist a large cross-section through A, even if the section does not have to be flat. For example, a variation on a result of T. Bartoszynski and L. Halbeisen Show moreBy the Kuratowski-Ulam theorem, if A subset of Rn+1 = R-n x R is a Borel set which has second category intersection with every ball (i.e., is "everywhere second category"), then there is a y is an element of R such that the section A boolean AND (R-n x {y}) is everywhere second category in R-n x {y}. If A is not Borel, then there may not exist a large cross-section through A, even if the section does not have to be flat. For example, a variation on a result of T. Bartoszynski and L. Halbeisen shows that there is an everywhere second category set A subset of Rn+1 such that for any polynomial p in n variables, A boolean AND graph(p) is finite. It is a classical result that under the Continuum Hypothesis, there is an everywhere second category set L in Rn+1 which has only countably many points in any first category set. In particular, L boolean AND graph(f) is countable for any continuous function f : R-n -> R. We prove that it is relatively consistent with ZFC that for any everywhere second category set A in Rn+1, there is a function f : R-n -> R which is the restriction to R-n of an entire function on C-n and is such that, relative to graph(f), the set A n graph(f) is everywhere second category. For any collection of less than 2(N)0 sets A, the function f can be chosen to work for all sets A in the collection simultaneously. Moreover, given a nonnegative integer k, a function g: R-n -> R of class C-k and a positive continuous function epsilon:R-n -> R, we may choose f so that for all multiindices alpha of order at most k and for all X is an element of R-n, vertical bar D-alpha g(x) - D(alpha)g(x)vertical bar Show less
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