[Paper Review] Partitions of unity

Jerzy Dydak|ArXiv.org|Oct 24, 2002

Homotopy and Cohomology in Algebraic Topology11 references16 citations

TL;DR

This paper introduces a novel framework for general topology using equicontinuous partitions of unity as a unifying tool, deriving classical results like the Tietze Extension Theorem, Stone's Theorem, and Tamano's Theorem in a unified manner. It establishes a calculus of partitions of unity that generalizes barycentric subdivision and provides a new algebraic approach to dimension theory and metric simplicial complexes.

ABSTRACT

The paper contains an exposition of part of topology using partitions of unity. The main idea is to create variants of the Tietze Extension Theorem and use them to derive classical theorems. This idea leads to a new result generalizing major results on paracompactness (Stone Theorem and Tamano Theorem), a result which serves as a connection to Ascoli Theorem. A new calculus of partitions of unity is introduced with applications to dimension theory and metric simplicial complexes. The geometric interpretation of this calculus is the barycentric subdivision of simplicial complexes. Also, joins of partitions of unity are often used; they are an algebraic version of joins of simplicial complexes.

Motivation & Objective

• To unify foundational concepts in general topology—normality, paracompactness, and the Tietze Extension Theorem—through the use of partitions of unity.
• To develop a new algebraic calculus of partitions of unity that generalizes geometric constructions such as barycentric subdivision.
• To establish a connection between partitions of unity and Ascoli-type theorems via equicontinuity.
• To provide a new characterization of paracompactness and normality using partitions of unity subordinated to open covers.
• To extend the applicability of partitions of unity beyond locally finite or point-finite cases by introducing equicontinuous families.

Proposed method

• Define normal and paracompact spaces via the existence of partitions of unity subordinated to finite open covers, replacing traditional definitions based on separation axioms.
• Use equicontinuous partitions of unity as a core tool to derive extensions of continuous functions, generalizing the Tietze Extension Theorem.
• Introduce a calculus of partitions of unity involving joins, which algebraically model joins of simplicial complexes.
• Apply the calculus to dimension theory, showing that partitions of unity can be used to define and compute dimension via refinement processes.
• Use inverse limits and compactifications to embed equicontinuous families of functions into continuous maps on product spaces with compact factors.
• Define strong equicontinuity via one-point compactification of the index set, linking continuity of the evaluation map to equicontinuity.

Experimental results

Research questions

• RQ1 Can partitions of unity be used to unify classical theorems in general topology, such as the Tietze Extension Theorem and Stone’s Theorem on paracompactness?
• RQ2 How can equicontinuous partitions of unity serve as a foundation for dimension theory and the theory of metric simplicial complexes?
• RQ3 What is the algebraic structure of joins of partitions of unity, and how does it relate to the geometric operation of barycentric subdivision?
• RQ4 Under what conditions can an equicontinuous family of functions on a subspace be extended to the whole space while preserving equicontinuity?
• RQ5 When does a family of functions on a space admit a continuous extension to a product space with a compact Hausdorff factor?

Key findings

• The paper establishes that a Hausdorff space is normal if and only if for every finite open cover, there exists a partition of unity subordinated to it, providing a new characterization of normality.
• A new generalization of the Tietze Extension Theorem is derived for equicontinuous partitions of unity, extending continuous functions from closed subsets to the whole space.
• The calculus of partitions of unity, including their joins, provides an algebraic model for barycentric subdivision in simplicial complexes.
• The paper proves that every equicontinuous family of functions from a space to a regular space, with pointwise precompact image, can be embedded into a continuous map from a product with a compact Hausdorff space.
• Strong equicontinuity of a family of functions is characterized by the continuity of the evaluation map on the product with the one-point compactification of the index set.
• The paper shows that a pointwise bounded equicontinuous family of real-valued functions on a subspace extends to the whole space if and only if the subspace is P-embedded, generalizing a result of Yamazaki.