The openness of induced maps on hyperspaces
Alejandro Illanes · Colloquium Mathematicum · 1998
A continuum is a compact connected metric space.A map is a continuous function.For a continuum X with metric d, C(X) denotes the hyperspace of subcontinua of X with the Hausdorff metric H.Given an onto map f : X → Y between continua, the induced mapProperties of induced maps have been studied by J. J. Charatonik, W. J. Charatonik and H. Hosokawa [2][3][4][5][6][7][8][9][10][11][12][13][14].In [13, Theorem 4.3], H. Hosokawa proved that if f 1 is open, then f is open and he gave an example showing that the converse of this implication is not true.In the same paper he asked the following question: Is there an open map f such that f 1 is open but f 2 is not open?In this paper we prove the following result.Theorem.Let f : X → Y be an onto map.If Y is nondegenerate and f 2 is open, then f is a homeomorphism.As a consequence of this result, we obtain a positive answer to Hosokawa's question.Concepts not defined here will be taken as they appear in [15].Lemma.Let f : X → Y be a confluent map, let x 0 ∈ X and let β be an order arc in C(Y ) such that f (x 0 ) ∈ B∈β B. Then there exists an order arc α in C(X) such that x 0 ∈ A∈α A and f 2 (α) = β.P r o o f.For each B ∈ β, let A B be the component of f -1 (B) such that x 0 ∈ A B , then f (A B ) = B. Define α 0 = {A B : B ∈ β}, B 0 = B∈β B and B 1 = B∈β B. Then α 0 has the following properties: (1) If A ∈ α 0 , then A B 0 ⊂ A ⊂ A B 1 and (2) If A 1 , A 2 ∈ α 0 , then A 1 ⊂ A 2 or A 2 ⊂ A 1 .