To any continuum X weassign an ordinal number (or the symbol ∞) s(X), called the degree of nonlocal connectedness of X. We show that (1) the degree cannot be increased under continuous surjections; (2) for hereditarily unicoherent continua X, the degree of a subcontinuum of X is less than or equal to s(X); (3) s(C(X)) ≤ s(X), where C(X) denotes the hyperspace of subcontinua of a continuum X. We also investigate the degrees of Cartesian products and inverse limits. As an application weconstruct an uncountable family of metric continua X homeomorphic to C(X).
J. J. Charatonik and W. J. Charatonik, "A Degree of Nonlocal Connectedness," Rocky Mountain Journal of Mathematics, Rocky Mountain Math Consortium, Jan 2001.
The definitive version is available at https://doi.org/10.1216/rmjm/1021249438
Mathematics and Statistics
Keywords and Phrases
Cartesian Product; Continuum; Degree; Hereditarily Unicoherent; Inverse Limit; Locally Connected; Ordinal Number; Hyperspace
International Standard Serial Number (ISSN)
Article - Journal
© 2001 Rocky Mountain Math Consortium, All rights reserved.
01 Jan 2001