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 Bulaʹs 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 Bulaʹs 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 Bulaʹs property. We construct another type of maps, namely, monotone open maps on n-manifolds, n≥3 with nontrivial fibers which do not have Bulaʹs property. Our construction essentially applies Brownʹs theorem (1958) on a continuous decomposition of Rn\{0} into hereditarily indecomposable continua separating between 0 and ∞. We present a relatively short proof of Brownʹs theorem based on the approach of Levin (1996). Related results are discussed.