[Paper Review] External characterization of I-favorable spaces

Vesko Valov|arXiv (Cornell University)|May 1, 2010

Advanced Banach Space Theory3 references3 citations

TL;DR

This paper provides a spectral and internal characterization of I-favorable spaces with respect to co-zero sets, establishing that such spaces are precisely those that are skeletally generated or π-regularly embedded in compact spaces. The key contribution is a complete equivalence between I-favorability, π-regular embeddability, and skeletally generated structure, with applications to products and C*-embedded subspaces of extremally disconnected spaces.

ABSTRACT

We provide both a spectral and an internal characterizations of arbitrary I-favorable spaces with respect to co-zero sets. As a corollary we establish that any product of compact I-favorable spaces with respect to co-zero sets is also I-favorable with respect to co-zero sets. We also prove that every C*-embedded I-favorable with respect to co-zero sets subspace of an extremally disconnected space is extremally disconnected.

Motivation & Objective

• To establish a complete external and internal characterization of I-favorable spaces with respect to co-zero sets.
• To prove that the class of I-favorable spaces with respect to co-zero sets coincides with skeletally generated spaces.
• To show that products of compact I-favorable spaces with respect to co-zero sets remain I-favorable.
• To demonstrate that any C*-embedded I-favorable subspace of an extremally disconnected space is itself extremally disconnected.

Proposed method

• Introduce a modified open-open game restricted to co-zero sets, defining I-favorability with respect to co-zero sets via a winning strategy for Player I.
• Define π-regular embeddability using a π-base and a function assigning open sets in the ambient space to co-zero sets in the subspace.
• Construct a σ-complete inverse system of separable metric spaces with skeletal bonding maps to define skeletally generated spaces.
• Prove equivalence between I-favorability, π-regular embeddability, and skeletally generated structure via a chain of implications.
• Use factorizability of σ-complete inverse systems with open projections and second-countable spaces to verify condition (7) in the skeletally generated definition.
• Apply results on inverse systems and continuous inverse limits to extend embeddings and preserve topological properties.

Experimental results

Research questions

• RQ1 What is the external characterization of I-favorable spaces with respect to co-zero sets?
• RQ2 How does I-favorability with respect to co-zero sets relate to π-regular embeddability in compact spaces?
• RQ3 Is the class of I-favorable spaces with respect to co-zero sets equivalent to skeletally generated spaces?
• RQ4 Are products of compact I-favorable spaces with respect to co-zero sets also I-favorable?
• RQ5 Does a C*-embedded I-favorable subspace of an extremally disconnected space inherit extremal disconnectedness?

Key findings

• A space is I-favorable with respect to co-zero sets if and only if it is π-regularly embedded in some compact space.
• A space is I-favorable with respect to co-zero sets if and only if it is skeletally generated via a σ-complete inverse system of separable metric spaces with skeletal bonding maps.
• Any product of compact I-favorable spaces with respect to co-zero sets is itself I-favorable with respect to co-zero sets.
• Every C*-embedded I-favorable subspace of an extremally disconnected space is extremally disconnected.
• The class of I-favorable spaces with respect to co-zero sets coincides with the class of skeletally generated spaces.
• The proof relies on the factorizability of σ-complete inverse systems with open projections and second-countable spaces, ensuring the existence of continuous lifts for bounded continuous functions.