Rosenthal compacta that are premetric of finite degree

Abstract

We show that if a separable Rosenthal compactum K is a continuous n-to-one preimage of a metric compactum, but it is not a continuous n − 1-to-one preimage, then K contains a closed subset homeomorphic to either the n−Split interval Sn(I) or the Alexandroff n−plicate Dn(2 N ). This generalizes a result of the third author that corresponds to the case n = 2.

Lexicographic products as compact spaces of the first Baire class

“…The two-to-one preimages of metric spaces (when C are the spaces of at most two points) are involved in several structural results for separable Rosenthal compacta [10], and some generalizations exist for finite-to-one preimages [2]. We will consider now the class of Corson compacta.…”

“…Given a class C of compact spaces, we say that a compact space K is a C-toone preimage of a metric space if there exists a continous function f : K −→ M onto a metric space M such that f −1 (x) ∈ C for every x ∈ M . The two-to-one preimages of metric spaces (when C are the spaces of at most two points) are involved in several structural results for separable Rosenthal compacta [10], and some generalizations exist for finite-to-one preimages [2]. We will consider now the class of Corson compacta.…”

Compact Spaces of the First Baire Class That Have Open Finite Degree

“…A compact space K is a premetric compactum of degree n if it is a premetric compactum of degree at most n but not a premetric compactum of at most n − 1. The main result of [3] is that, for every n, there exist two Rosenthal premetric compacta S n (I ) and D n (2 N ) of degree n such that every separable Rosenthal premetric compactum of degree n contains a homeomorphic copy of either S n (I ) or D n (2 N ). Although there is some superficial similarity, the result from [3] is not deduced from the results of this paper, nor vice versa.…”

“…On the other hand, L is a premetric compactum of degree m − 1. In fact f (x,P) : x ∈ 2 ω , P ∈ P is homeomorphic to the space D m−1 (2 N ) of [3]. for all z ∈ n ω and all i < j < n.…”

“…Section 7 studies the open degree, and another related degree. It is proven in [2] that, for this other notion of degree, every separable Rosenthal compact space of degree n contains one of two basic spaces. The differences with the results of this paper are discussed.…”