Topology Proceedings 33 (2009), pp. 251-268

Recognizing indecomposable subcontinua of surfaces from their complements

Clinton P. Curry

We prove two theorems which allow one to recognize indecomposable subcontinua of closed surfaces without boundary. If X is a subcontinuum of a closed surface S, we call the components of S\X the complementary domains of X. We prove that a continuum X is either indecomposable or the union of two indecomposable continua whenever it has a sequence (U_n)_n=1^∞ of distinct complementary domains such that lim_{n → ∞} ∂U_n = X. We define a slightly stronger condition on the complementary domains of X, called the double-pass condition, which we conjecture is equivalent to indecomposability. We prove that this is so for continua which are not the boundary of one of their complementary domains.

Keywords: indecomposable continuum, complementary domain, closed surface, double-pass condition

Mathematics Subject Classification: 54F15 (57N05)

Document formats: PDF file 197.5 Kb

Topology Proceedings, Volume 33 (2009)
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