Uncountable absorbing systems related to the Hausdorff dimension

N.Mazurenko

Given any $n\in\mathbb N$, we describe the topology of the system $\{HD_{>\gamma}({\mathbb I}^n)\}_{\gamma\in[0,n)}$ consisting of spaces of compact subsets of Hausdorff dimension $>\gamma$ in the $n$-dimensional cube $\mathbb I^n$.

1. T.~Banakh, T.~Radul, M.~Zarichnyi, Absorbing Sets in Infinite-Dimensional Manifolds. -- Lviv: VNTL Publishers, 1996. -- Vol. 1. -- 231 p.

2. C.~Bessaga, A. Peczyński, Selected topics in infinite-dimensional topology. -- Warszawa, 2001. -- 352 p.

3. D.W.~Curtis, R.M.~Schori, Hyperspaces of Peano continua are Hilbert cubes, Fund. Math. 101 (1978) №1, 19--38.

4. D.W.~Curtis, Hyperspaces homeomorphic to Hilbert space, Proc. Amer. Math. Soc. 75 (1979) №1, 126--130.

5. R.~Cauty R, Suites ${\mathcal F}_\sigma$-absorbantes en theorie de la dimension, Fund. Math. 159 (1999) №2, 115--126.

6. J.J.~Dijkstra, J.~van Mill, J.~Mogilski, The space of infinite-dimensional compacta and other topological copies of $(l^2_f)^\omega$, Pacif. J. Math. 152 (1992) 255--273.

7. T.~Dobrowolski, L.R.~Rubin, The hyperspace of infinite-dimensional compacta for covering and cohomological dimension are homeomorphic, Pacific J. Math. 164 (1994) №1, 15--39.

8. G.A.~Edgar, Measure, Topology and Fractal Geometry. -- New York: Springer-Verlag, 1995. -- 221 p.

9. K.J.~Falconer K, The Geometry of Fractal Sets. -- Cambridge University Press, 1985. -- 162 p.

10. H.~Gladdines, Absorbing systems in infinite-dimensional manifolds and applications. -- Amsterdam: Vrije Universiteit, 1994. -- 117p.

11. N.~Mazurenko, Absorbing sets related to Hausdorff dimension, Visnyk Lviv Univ., Ser. Mech-Math. 61 (2003) 121--128.

12. J.~van Mill, Infinite-Dimensional Topology. -- Amsterdam: Vrije Universiteit, 1989. -- 401 p.

13. J.~van Mill, The Infinite-Dimensional Topology of Function Spaces. -- Amsterdam: Elsevier, 2001. -- Volume 64. -- 630p.

14. Н.~Мазуренко, Топологія гіперпросторів континуумів заданого виміру Гаусдорфа в скінченновимірному кубі, Наук. Вісник Чернівецького унів. Математика. 228 (2004) 60--65.

	Uncountable absorbing systems related to the Hausdorff dimension
Author	N.Mazurenko mnatali@ukr.net Precarpatian National University,Ivano-Frankivsk, Ukraine
Abstract	Given any $n\in\mathbb N$, we describe the topology of the system $\{HD_{>\gamma}({\mathbb I}^n)\}_{\gamma\in[0,n)}$ consisting of spaces of compact subsets of Hausdorff dimension $>\gamma$ in the $n$-dimensional cube $\mathbb I^n$.
Keywords	uncountable absorbing system, Hausdorff dimension, compact subset
DOI	doi:10.30970/ms.31.2.195-203
Reference	1. T.~Banakh, T.~Radul, M.~Zarichnyi, Absorbing Sets in Infinite-Dimensional Manifolds. -- Lviv: VNTL Publishers, 1996. -- Vol. 1. -- 231 p. 2. C.~Bessaga, A. Peczyński, Selected topics in infinite-dimensional topology. -- Warszawa, 2001. -- 352 p. 3. D.W.~Curtis, R.M.~Schori, Hyperspaces of Peano continua are Hilbert cubes, Fund. Math. 101 (1978) №1, 19--38. 4. D.W.~Curtis, Hyperspaces homeomorphic to Hilbert space, Proc. Amer. Math. Soc. 75 (1979) №1, 126--130. 5. R.~Cauty R, Suites ${\mathcal F}_\sigma$-absorbantes en theorie de la dimension, Fund. Math. 159 (1999) №2, 115--126. 6. J.J.~Dijkstra, J.~van Mill, J.~Mogilski, The space of infinite-dimensional compacta and other topological copies of $(l^2_f)^\omega$, Pacif. J. Math. 152 (1992) 255--273. 7. T.~Dobrowolski, L.R.~Rubin, The hyperspace of infinite-dimensional compacta for covering and cohomological dimension are homeomorphic, Pacific J. Math. 164 (1994) №1, 15--39. 8. G.A.~Edgar, Measure, Topology and Fractal Geometry. -- New York: Springer-Verlag, 1995. -- 221 p. 9. K.J.~Falconer K, The Geometry of Fractal Sets. -- Cambridge University Press, 1985. -- 162 p. 10. H.~Gladdines, Absorbing systems in infinite-dimensional manifolds and applications. -- Amsterdam: Vrije Universiteit, 1994. -- 117p. 11. N.~Mazurenko, Absorbing sets related to Hausdorff dimension, Visnyk Lviv Univ., Ser. Mech-Math. 61 (2003) 121--128. 12. J.~van Mill, Infinite-Dimensional Topology. -- Amsterdam: Vrije Universiteit, 1989. -- 401 p. 13. J.~van Mill, The Infinite-Dimensional Topology of Function Spaces. -- Amsterdam: Elsevier, 2001. -- Volume 64. -- 630p. 14. Н.~Мазуренко, Топологія гіперпросторів континуумів заданого виміру Гаусдорфа в скінченновимірному кубі, Наук. Вісник Чернівецького унів. Математика. 228 (2004) 60--65.
Pages	195-203
Volume	31
Issue	2
Year	2009
Journal	Matematychni Studii
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