Open maps on manifolds which do not admit disjoint closed subsets intersecting each fiber

Let X and Y be compacta. A map f : X → Y is said to satisfy Bula's property if there exist disjoint closed subsets F₀ and F₁ of X such that f(F₀) = f(F₁) = 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.