TY - JOUR

T1 - The space of composants of an indecomposable continuum

AU - Solecki, Sławomir

N1 - Funding Information:
1Research supported by NSF Grant DMS-9803676. Current address: Department of Mathematics, University of Illinois, 1409 W. Green Street, Urbana, IL 61801. 2For terminology unexplained in 1.1–1.3 see 1.4.

PY - 2002/3/25

Y1 - 2002/3/25

N2 - The family of all composants of an indecomposable continuum is studied. We investigate the equivalence relation induced on an indecomposable continuum by its partition into composants. We show that up to Borel bireducibility such an equivalence relation can be of only two types: E0 and E1, the E0 type being "simple" and the E1 type being "complicated." As a consequence of this, we show that each indecomposable continuum carries a Borel probability measure which assigns 0 to each composant and 0 or 1 to each Borel set which is the union of a family of composants. In particular, it follows that there is no Borel transversal for the family of all composants. This solves an old problem in the theory of continua. We prove that all hereditarily indecomposable continua are of the complicated type, that is, they fall into the E1 type. We analyze the properties of being of type E1 or of type E0. We show, using effective descriptive set theory, that the first of these properties is analytic and so the second one is coanalytic. We construct examples of continua of both types; in fact, we produce a family of indecomposable continua and use it to prove that these properties are complete analytic and complete coanalytic, respectively, hence non-Borel, so they do not admit simple topological characterizations. We also use continua from this family to show that an indecomposable continuum may be of type E1 only because of the behavior of composants on a small subset of the continuum. This, in particular, shows that certain natural approaches to solving Kuratowski's problem on generic ergodicity of the composant equivalence relation will not work. We finish with some open problems.

AB - The family of all composants of an indecomposable continuum is studied. We investigate the equivalence relation induced on an indecomposable continuum by its partition into composants. We show that up to Borel bireducibility such an equivalence relation can be of only two types: E0 and E1, the E0 type being "simple" and the E1 type being "complicated." As a consequence of this, we show that each indecomposable continuum carries a Borel probability measure which assigns 0 to each composant and 0 or 1 to each Borel set which is the union of a family of composants. In particular, it follows that there is no Borel transversal for the family of all composants. This solves an old problem in the theory of continua. We prove that all hereditarily indecomposable continua are of the complicated type, that is, they fall into the E1 type. We analyze the properties of being of type E1 or of type E0. We show, using effective descriptive set theory, that the first of these properties is analytic and so the second one is coanalytic. We construct examples of continua of both types; in fact, we produce a family of indecomposable continua and use it to prove that these properties are complete analytic and complete coanalytic, respectively, hence non-Borel, so they do not admit simple topological characterizations. We also use continua from this family to show that an indecomposable continuum may be of type E1 only because of the behavior of composants on a small subset of the continuum. This, in particular, shows that certain natural approaches to solving Kuratowski's problem on generic ergodicity of the composant equivalence relation will not work. We finish with some open problems.

KW - Borel equivalence relations

KW - Composants

KW - Indecomposable continua

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U2 - 10.1006/aima.2001.2002

DO - 10.1006/aima.2001.2002

M3 - Article

AN - SCOPUS:0037171109

SN - 0001-8708

VL - 166

SP - 149

EP - 192

JO - Advances in Mathematics

JF - Advances in Mathematics

IS - 2

ER -