A well-quasi-order for continuous functions

TL;DR

This paper establishes that continuous reducibility forms a well-quasi-order among continuous functions between certain topological spaces, introducing the concept of scattered functions to classify these functions up to continuous equivalence.

Contribution

It proves the well-quasi-order property for continuous reducibility and introduces scattered functions as a new classification tool for continuous functions.

Findings

Continuous reducibility is a well-quasi-order on the specified class of functions.

Scattered functions generalize scattered spaces and are exhaustively classified.

The paper characterizes functions up to continuous equivalence using scattered functions.

Abstract

We prove that continuous reducibility is a well-quasi-order on the class of continuous functions between separable metrizable spaces with analytic zero-dimensional domain. To achieve this, we define scattered functions, which generalize scattered spaces, and describe exhaustively scattered functions between zero-dimensional separable metrizable spaces up to continuous equivalence.