Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
|
   
Available in electronic format
Available in print format 		Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(e) ISSN 0002-9947(p)

     

On the dimensional structure of hereditarily indecomposable continua

Author(s): Roman Pol; Miroslawa Renska
Journal: Trans. Amer. Math. Soc. 354 (2002), 2921-2932.
MSC (1991): Primary 54F15, 54F45, 54H05
Posted: March 6, 2002
MathSciNet review: 1895209
Retrieve article in: PDF
This article is available free of charge

Abstract | References | Similar articles | Additional information

Abstract: Any hereditarily indecomposable continuum $X$ of dimension $n\geq 2$ is split into layers $B_r$ consisting of all points in $X$ that belong to some $r$-dimensional continuum but avoid any non-trivial continuum of dimension less than $r$. The subjects of this paper are the dimensional and the descriptive properties of the layers $B_r$.

References:

[Al1]
P.S.Alexandroff, On some main directions in general topology, Uspekhi Mat. Nauk 19 (1964), 3-46 (in Russian).

[Al2]
P.S.Alexandroff, Introduction to Homological Dimension Theory, ``Nauka", Moscow, 1975 (in Russian).

[Bi]
R.H.Bing, Higher-dimensional hereditarily indecomposable continua, Trans. Amer. Math. Soc. 71 (1951), 267-273. MR 13:265c

[Ch-P]
V.A.Chatyrko, E.Pol, Continuum many Fréchet types of hereditarily strongly infinite-dimensional Cantor manifolds, Proc. Amer. Math. Soc. 128 (2000), 1207 - 1213. MR 2000i:54051

[En]
R.Engelking. Theory of Dimensions, Finite and Infinite, Heldermann Verlag, Lemgo, 1995. MR 97j:54033

[Ho]
W.Holsztynski, Universality of the product mappings onto products of $I^n$ and snake-like spaces, Fund. Math. 64 (1969), 147-155. MR 39:6249

[Kar]
Z.Karno, On a theorem of P.S. Aleksandrov, Coll. Math. 72 (1997), 39-51. MR 98b:54047

[Kau1]
B.Kaufmann, The dissection of closed sets of arbitrary dimension and the generalized Brouwer-Alexandroff theorem, Proc. Camb. Phil. Soc. 31 (1935), 525-535.

[Kau2]
B.Kaufmann, On infinitesimal properties of closed sets of arbitrary dimension, Annals of Math. 38 (1937), 14-35

[Kau3]
B.Kaufmann, Of the extension of the Pflastersatz, Proc. Camb. Phil. Soc. 32 (1936), 238-247.

[Kel]
J.L.Kelley, Hyperspaces of a continuum, Trans. Amer. Math. Soc. 52 (1942), 22-36. MR 3:315b

[Kn]
B.Knaster, Sur les coupures biconnexes des espaces euclidiens de dimension $n>1$ arbitraire, Matem. Sbornik 19 (1946), 9-18. MR 8:164j

[Kra1]
J.Krasinkiewicz, Homotopy separators and mappings into cubes, Fund. Math. 131 (1988), 149-154. MR 90i:54081

[Kra2]
J.Krasinkiewicz, On mappings with hereditarily indecomposable fibers, Bull. Pol. Acad. Sci. 44 (1996), 147-156. MR 97g:54016

[Kra3]
J.Krasinkiewicz, On approximation of mappings into 1-manifolds, Bull. Pol. Acad. Sci. 44 (1996), 431-440. MR 97k:54011

[Kur]
K.Kuratowski, Topology, vol. I, Academic Press, New York, 1966, and vol. II, Academic Press, New York, 1968. MR 36:840; MR 41:4467

[Lev]
M.Levin, Bing maps and finite-dimensional maps, Fund. Math. 151 (1996), 47-52. MR 97e:54031

[L-S]
M.Levin, Y.Sternfeld, Hyperspaces of two-dimensional continua, Fund. Math. 150 (1996), 17-24. MR 97d:54015

[Lew]
W.Lewis, The pseudo-arc, Bol. Soc. Mat. Mexicana 5 (1999), 25-77. MR 2000f:54029

[Ma]
S.Mazurkiewicz, Sur les problemes $\kappa$ and $\lambda$ de Urysohn, Fund. Math. 10 (1926), 311-319.

[vM-P]
J.van Mill and R.Pol, On the existence of weakly $n$-dimensional spaces, Proc. Amer. Math. Soc. 113 (1991), 581-585. MR 91m:54043

[Mor]
K.Morita, Cech cohomology and covering dimension for topological spaces, Fund. Math. 87 (1975), 31-52. MR 50:14706

[P-R]
E.Pol, M.Renska, On Bing points in infinite-dimensional hereditarily indecomposable continua, Topology and Appl. (to appear).

[Po1]
R.Pol, Countable dimensional universal sets, Trans. Amer. Math. Soc. 297 (1986), 255-268. MR 87h:54067

[Po2]
R.Pol, An $n$-dimensional compactum which remains $n$-dimensional after removing all Cantor $n$-manifolds, Fund. Math. 136 (1990), 127-131. MR 91j:54061

[Ro]
J.T.Rogers, Jr., Orbits of higher-dimensional hereditarily indecomposable continua, Proc. Amer. Math. Soc. 95 (1985), 483-485. MR 86k:54054

[R-S-W]
L.R.Rubin, R.M.Schori, and J.J.Walsh, New dimension theory technique for constructing infinite-dimensional examples, General Topology Appl. 10 (1979), 93-102. MR 80e:54049

[St]
Y.Sternfeld, On finite-dimensional maps and other maps with ``small" fibres, Fund. Math. 147 (1995), 127-133. MR 96e:54030

Similar Articles:

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 54F15, 54F45, 54H05

Retrieve articles in all Journals with MSC (1991): 54F15, 54F45, 54H05

Additional Information:

Roman Pol
Affiliation: Institute of Mathematics, University of Warsaw, Banacha 2, 02-097 Warsaw, Poland
Email: pol@mimuw.edu.pl

Miroslawa Renska
Affiliation: Institute of Mathematics, University of Warsaw, Banacha 2, 02-097 Warsaw, Poland
Email: mrenska@mimuw.edu.pl

DOI: 10.1090/S0002-9947-02-02959-8
PII: S 0002-9947(02)02959-8
Keywords: Hereditarily indecomposable continua, dimension, Borel sets
Received by editor(s): September 5, 2000
Received by editor(s) in revised form: October 5, 2001
Posted: March 6, 2002
Additional Notes: Research partially supported by KBN grant 5 P03A 024 20
Copyright of article: Copyright 2002, American Mathematical Society




AMS and Social Media LinkedIn Facebook Podcasts Twitter YouTube RSS Feeds Blogs Wikipedia