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In 1977 T. Maćkowiak proved that each local homeomorphism from a continuum onto a tree-like continuum is a homeomorphism. Recently, J. Rogers proved that each locally one-to-one (not necessarily open) map from a hereditarily decomposable continuum onto a tree-like continuum is a homeomorphism, and this paper removes “hereditarily decomposable” from the hypothesis of Rogers’ theorem. It is not easy for a nice function to map onto a tree-like continuum without being a homeomorphism. T. Maćkowiak’s classic result [3], proved in 1977, is that each local homeomorphism from a continuum onto a tree-like continuum is a homeomorphism. Local homeomorphisms are both open and locally one-to-one, and recently J. Rogers asked if “open” could be removed from the hypothesis of the Maćkowiak theorem. In [1] Rogers proved a special case, namely that if a locally one-to-one map that is not a homeomorphism is defined on a hereditarily decomposable continuum, then the image contains a continuum that is not unicoherent. Since all tree-like continua are hereditarily unicoherent, the image cannot be tree-like. These results come from the “complicated proof” found in [2] of Corollary 5.7 in [1]. We use the noun “map” to mean continuous function, and the term “continuum” to mean a connected, compact metric space. The theorem to follow completes the task of removing “open” from the Maćkowiak theorem. The lemma that is proved first helps to organize the covers. Definition. A finite collection of sets has a tree-indexing if its members can be labeled {L1, L2, ..., Lm} so that the Li are distinct and for each j from 2 to m, Lj intersects exactly one member of the set {L1, L2, ..., Lj−1}. Tree-cover lemma. A finite collection of open sets has a tree-indexing iff its nerve is a tree. Proof. Suppose {L1, L2, ..., Lm} is a tree-indexing of a finite collection V of open sets. Then V is coherent since if V = V1 ∪ V2, two disjoint subcollections whose unions do not intersect, and if L1 ∈ V1, then the element of V2 with the smallest label fails the definition of a tree-indexing. Since V is coherent, its nerve is connected. Since no three elements in V can intersect, its nerve is a one-dimensional graph. If the nerve of V contains a simple closed curve, then the element of V with the Received by the editors January 30, 1995. 1991 Mathematics Subject Classification. Primary 54C10.

Each locally one-to-one map from a continuum onto a tree-like continuum is a homeomorphism