Transfinite Sequences of Continuous and Baire Class 1 Functions

TL;DR

This paper explores how the length of well-ordered monotone sequences of continuous and Baire class 1 functions varies depending on the underlying metric space, revealing insights into their structural properties.

Contribution

It characterizes the possible lengths of such sequences in relation to different metric spaces, advancing understanding of their order-theoretic structure.

Findings

Lengths depend on the properties of the metric space

Certain spaces admit arbitrarily long well-ordered sequences

Results connect topological features with order-theoretic properties

Abstract

The set of continuous or Baire class 1 functions defined on a metric space $X$ is endowed with the natural pointwise partial order. We investigate how the possible lengths of well-ordered monotone sequences (with respect to this order) depend on the space $X$.