Most Cantor sets in RN are in general position with respect to all projections

• Autorzy: Frolkina Olga

Identyfikatory

DOI

10.4064/ba221106-7-11

• Języki publikacji: EN

Abstrakty

EN

We prove the theorem stated in the title. This answers a question of John Cobb (1994). We also consider the case of the Hilbert space ℓ2.

Słowa kluczowe

EN

Euclidean space; Hilbert space; projection; Cantor set; dimension; general position; Baire Category Theorem

• Wydawca: Instytut Matematyczny PAN
• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2022
• Tom: Vol. 70, no. 1
• Strony: 53--62
• Opis fizyczny: Bibliogr. 14 poz.

Twórcy

autor

Frolkina Olga

• https://orcid.org/0000-0001-8680-3952

Bibliografia

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• [3] K. Borsuk, An example of a simple arc in space whose projection in every plane has interior points, Fund. Math. 34 (1947), 272-277.
• [4] J. Cobb, Raising dimension under all projections, Fund. Math. 144 (1994), 119-128.
• [5] J. J. Dijkstra and J. van Mill, Projections of planar Cantor sets in potential theory, Indag. Math. (N.S.) 8 (1997), 173-180.
• [6] O. Frolkina, A Cantor set in Rd with “large” projections, Topology Appl. 157 (2010), 745-751.
• [7] O. Frolkina, Cantor sets with high-dimensional projections, Topology Appl. 275 (2020), art. 107020, 13 pp.
• [8] O. Frolkina, All projections of a typical Cantor set are Cantor sets, Topology Appl. 281 (2020), art. 107192, 11 pp.
• [9] O. Frolkina, A new simple family of Cantor sets in R3 all of whose projections are one-dimensional, Topology Appl. 288 (2021), art. 107452, 11 pp.
• [10] P. Gartside and M. Kovan-Bakan, On the space of Cantor subsets of R3, Topology Appl. 160 (2013), 1088-1098.
• [11] A. Kechris, Classical Descriptive Set Theory, Springer, 1995.
• [12] K. Kuratowski, Applications of the Baire-category method to the problem of independent sets, Fund. Math. 81 (1973), 65-72.
• [13] A. V. Kuz’minykh, The structure of typical compact sets in Euclidean space, Siberian Math. J. 38 (1997), 269-301.
• [14] J. Mycielski, Almost every function is independent, Fund. Math. 81 (1973), 43-48.

Uwagi

Opracowanie rekordu ze środków MEiN, umowa nr SONP/SP/546092/2022 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2022-2023).