Cleavability and scattered sets of non-trivial fibers

In this paper, we provide a partial answer to a problem posed by A. V.Arhangel’skii; we show that if X is a compactum cleavable over a separable linearly ordered topological space (LOTS) Y such that for some continuous function f from X to Y, the set of points on which f is not injective is scattered, then X is a LOTS.

💡 Research Summary

The paper addresses a classical problem in topology concerning “cleavability,” a notion introduced by Arhangel’skiĭ and Shakhmatov. A space X is said to be cleavable over a space Y if for every subset A⊆X there exists a continuous map f:X→Y such that the images of A and its complement are disjoint. Two central questions have guided the literature: (1) does cleavability of X over a Hausdorff space Y force X to embed as a subspace of Y? (2) if an infinite compact space X is cleavable over a linearly ordered topological space (LOTS), must X itself be a LOTS?

The author provides a partial affirmative answer to the second question under an additional hypothesis concerning the set of non‑injectivity points of a particular continuous map. Specifically, let f:X→Y be a continuous function and define M_f = {x∈X : ∃x’≠x with f(x)=f(x’)}. The main theorem asserts: If X is a compact space cleavable over a separable LOTS Y, and there exists a continuous f for which M_f is scattered, then X is itself a LOTS.

The proof is divided into two major parts.

1. The totally disconnected case (Section 2).
Here X is assumed to be zero‑dimensional, compact, and T₂. The author first establishes Lemma 2.7, which provides a systematic way to partition any countable closed subset A⊆X (homeomorphic to a countable ordinal) into pairwise disjoint clopen neighborhoods U_α, each containing a distinct point x_α∈A. The construction respects arbitrary clopen sub‑collections, guaranteeing that unions of selected U_α together with the corresponding points of A remain clopen.

Lemma 2.8 builds on this partition. Given a point x∈M_f, the lemma constructs a new separable LOTS Y₁ and a continuous map f₁:X→Y₁ such that x is no longer a point of non‑injectivity (i.e., x∉M_{f₁}) while the rest of M_f is preserved (M_{f₁}⊂M_f). The idea is to view the fiber A=f⁻¹(f(x)) as a countable ordinal λ, embed λ into a new linear order, and “stretch” the original Y by inserting isolated copies of the ordinals corresponding to the clopen neighborhoods U_α. This yields a larger ordered space that has enough “room” to separate the points of the fiber.

Lemmas 2.9 and 2.10 show that if M_f is scattered then its image f(M_f) is also scattered and has Cantor–Bendixson rank < ω₁. This rank bound is crucial because it permits a transfinite induction on the rank of f(M_f).

Theorem 2.11 then proceeds by induction. The base case (rank 0) is trivial. For rank 1, each isolated point y∈f(M_f) is surrounded by a clopen interval U_y; Lemma 2.7 supplies the required clopen neighborhoods, and Lemma 2.8 replaces each U_y by a new LOTS where the fiber over y becomes injective. The induction step assumes the result for all ranks < α and treats the α‑th derived set of f(M_f) similarly, using the previously constructed partitions to keep the process coherent across the whole space. After finitely many steps (since the rank is countable), a continuous map g:X→Z with empty M_g is obtained. Because g is injective and X is compact, g is a topological embedding; Z is a LOTS, so X is a closed subspace of a LOTS, which forces X itself to be a LOTS.

2. The general compact case (Section 3).
When X is not assumed totally disconnected, the author reduces the problem to the previous case. By separating X into its totally disconnected component and the remainder, and applying the results of Section 2 to each piece, the same inductive scheme works. The scatteredness of M_f guarantees that each non‑injective fiber can be isolated and treated independently.

Significance.
The paper shows that the combination of cleavability over a separable LOTS and the scatteredness of a single non‑injective fiber set is enough to force a compact space to inherit a linear order topology. This bridges a gap between the abstract splitting property and concrete orderability, providing a positive answer to Arhangel’skiĭ’s Question 2 in a broad class of spaces. The method highlights the power of Cantor–Bendixson analysis together with careful construction of auxiliary ordered spaces, and suggests possible extensions to non‑compact or non‑separable settings.