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Some complexity results in topology and analysis

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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 $Σ_2^1$ or PCA set. We show (a) there is an n-dimensional continuum X in $ℝ^n+1$ for which K(X) is a complete $Π_1^1$ set. In particular, $K(X) ∈ Π_1^1-Σ_1^1$; K(X) is coanalytic but is not an analytic set and (b) there is an n-dimensional continuum X in $ℝ^n+2$ for which K(X) is a complete $Σ_2^1$ set. In particular, $K(X) ∈ Σ_2^1-Π_2^1$; 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.
Twórcy
  • Department of Mathematics, University of North Texas, Denton, Texas 76203, U.S.A.
Bibliografia
  • [A] P. S. Aleksandrov, On the dimension of closed sets, Uspekhi Mat. Nauk 4 (6) (1949), 17-88 (in Russian).
  • S. I. Adyan and P. S. Novikov, On a semicontinuous function, Moskov. Gos. Ped. Inst. Uchen. Zap. 138 (3) (1958), 3-10 (in Russian).
  • [B] B. L. Brechner, On the dimensions of certain spaces of homeomorphisms, Trans. Amer. Math. Soc. 121 (1966), 516-548.
  • [E] R. Engelking, Dimension Theory, PWN and North-Holland, Warszawa-Amsterdam 1978.
  • [M] J. van Mill, n-dimensional totally disconnected topological groups, Math. Japon. 32 (1987), 267-273.
  • [P] R. Pol, An n-dimensional compactum which remains n-dimensional after removing all Cantor n-manifolds, Fund. Math. 136 (1990), 127-131.
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bwmeta1.element.bwnjournal-article-fmv141i1p75bwm
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