# Some complexity results in topology and analysis

Fundamenta Mathematicae (1992)

- Volume: 141, Issue: 1, page 75-83
- ISSN: 0016-2736

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topJackson, Steve, and Mauldin, R.. "Some complexity results in topology and analysis." Fundamenta Mathematicae 141.1 (1992): 75-83. <http://eudml.org/doc/211952>.

@article{Jackson1992,

abstract = {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.},

author = {Jackson, Steve, Mauldin, R.},

journal = {Fundamenta Mathematicae},

keywords = {cantor manifold; dimensional kernel; projective set; countably continuous; upper semicontinuous; -dimensional Cantor manifold; descriptive complexity},

language = {eng},

number = {1},

pages = {75-83},

title = {Some complexity results in topology and analysis},

url = {http://eudml.org/doc/211952},

volume = {141},

year = {1992},

}

TY - JOUR

AU - Jackson, Steve

AU - Mauldin, R.

TI - Some complexity results in topology and analysis

JO - Fundamenta Mathematicae

PY - 1992

VL - 141

IS - 1

SP - 75

EP - 83

AB - 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.

LA - eng

KW - cantor manifold; dimensional kernel; projective set; countably continuous; upper semicontinuous; -dimensional Cantor manifold; descriptive complexity

UR - http://eudml.org/doc/211952

ER -

## References

top- [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). Zbl1296.26015
- [B] B. L. Brechner, On the dimensions of certain spaces of homeomorphisms, Trans. Amer. Math. Soc. 121 (1966), 516-548. Zbl0151.30601
- [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. Zbl0622.22003
- [P] R. Pol, An n-dimensional compactum which remains n-dimensional after removing all Cantor n-manifolds, Fund. Math. 136 (1990), 127-131. Zbl0713.54039

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