[Paper Review] On hereditarily indecomposable compacta and factorization of maps

Klaas Pieter Hart, Elżbieta Pol|arXiv (Cornell University)|May 15, 2008

Advanced Topology and Set Theory6 references3 citations

TL;DR

This paper establishes a general factorization theorem for perfect maps with hereditarily indecomposable fibers, using model-theoretic techniques and the Löwenheim-Skolem principle to extend such maps to hereditarily indecomposable compactifications. The key contribution is a new proof of Maćkowiak's theorem on universal hereditarily indecomposable continua via Čech-Stone compactifications and pseudosuspension methods.

ABSTRACT

We prove a general factorization theorem for maps with hereditarily indecomposable fibers and apply it to reprove a theorem of Mackoviak on the existence of universal hereditarily indecomposable continua.

Motivation & Objective

• To establish a general factorization theorem for perfect maps with hereditarily indecomposable fibers in separable metrizable spaces.
• To reprove Maćkoviak's existence theorem on universal hereditarily indecomposable continua using a novel topological and model-theoretic approach.
• To demonstrate that the Čech-Stone compactification of a hereditarily indecomposable space remains hereditarily indecomposable.
• To show that every hereditarily indecomposable compactum of weight τ and dimension n admits a universal hereditarily indecomposable compactification of the same weight and dimension.
• To extend the applicability of factorization techniques to inverse limits and compactifications via elementary sublattices of closed sets.

Proposed method

• Utilizes Wallman’s representation theorem to represent compact Hausdorff spaces as spectra of lattices of closed sets.
• Applies a dual version of the Löwenheim-Skolem theorem from model theory to construct elementary sublattices of closed sets that preserve Property (KM).
• Employs Mardešić’s Factorization Theorem in a strengthened form to factor maps through hereditarily indecomposable compactifications.
• Uses the pseudosuspension construction to lift compactifications of zero-dimensional spaces into hereditarily indecomposable continua.
• Applies the property (KM) as a lattice-theoretic condition equivalent to hereditary indecomposability, enabling transfer of structure to compactifications.
• Constructs universal compacta via inverse systems of metrizable hereditarily indecomposable spaces indexed by cardinals of weight τ.

Experimental results

Research questions

• RQ1 Can perfect maps with hereditarily indecomposable fibers be factored through hereditarily indecomposable compactifications?
• RQ2 Does the Čech-Stone compactification of a hereditarily indecomposable space remain hereditarily indecomposable?
• RQ3 Can a universal hereditarily indecomposable compactum of given weight and dimension be constructed for any cardinal τ and dimension n?
• RQ4 To what extent can model-theoretic techniques, such as elementary sublattices, be used to preserve topological properties in compactifications?
• RQ5 Is there a factorization method that allows the construction of universal hereditarily indecomposable continua via pseudosuspension and inverse limits?

Key findings

• Every perfect map f:X→Y with hereditarily indecomposable fibers from a separable metrizable space X to a zero-dimensional separable metrizable space Y extends to a map f*:X*→Y* between hereditarily indecomposable and zero-dimensional metrizable compactifications.
• The Čech-Stone compactification βX of a hereditarily indecomposable space X is itself hereditarily indecomposable, provided X satisfies Property (KM).
• For every cardinal τ and n∈{0,1,…,∞}, there exists a hereditarily indecomposable compactum X(n,τ) of weight τ and dimension n that contains a copy of every hereditarily indecomposable compactum of weight ≤τ and dimension ≤n.
• Every hereditarily indecomposable compact space X of dimension n and weight τ admits an inverse system of metrizable hereditarily indecomposable compacta of dimension n and cardinality ≤τ whose limit is homeomorphic to X.
• Every normal n-dimensional space of weight τ with Property (KM) admits a hereditarily indecomposable compactification of dimension n and weight τ.
• The results extend to completely regular spaces when Property (KM) is reformulated using zero-sets and cozero-sets instead of closed and open sets.