## Abstract

Let X and Y be compacta. A map f : X → Y is said to satisfy Bula's property if there exist disjoint closed subsets F_{0} and F_{1} of X such that f(F_{0}) = f(F_{1}) = Y. It is well known that a surjective open map f : X → Y with infinite fibers satisfies Bula's property provided Y is finite-dimensional. In 1990 Dranishnikov constructed an open surjective map of infinite-dimensional compacta with fibers homeomorphic to a Cantor set which does not satisfy Bula's property. We construct another type of maps, namely, monotone open maps on n-manifolds, n ≥ 3 with nontrivial fibers which do not have Bula's property. Our construction essentially applies Brown's theorem (1958) on a continuous decomposition of Rdbl;^{n} \ {0} into hereditarily indecomposable continua separating between 0 and ∞. We present a relatively short proof of Brown's theorem based on the approach of Levin (1996). Related results are discussed.

Original language | English |
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Pages (from-to) | 221-228 |

Number of pages | 8 |

Journal | Topology and its Applications |

Volume | 103 |

Issue number | 2 |

DOIs | |

State | Published - 1 Jan 2000 |

Externally published | Yes |

## Keywords

- Dimension
- Hereditarily indecomposable continua
- Manifolds
- Open maps

## ASJC Scopus subject areas

- Geometry and Topology