Classification of spaces of Continuous Function on ordinals

TL;DR

This paper completes the topological classification of spaces of continuous real-valued functions on ordinal segments, showing their topological and uniform classifications coincide, thus advancing understanding of these function spaces.

Contribution

It finalizes the classification of $C_p([0,eta])$ spaces on ordinals, extending and completing R. Gorak's previous work.

Findings

Topological and uniform classifications of $C_p([0,eta])$ spaces coincide.

Complete classification of continuous function spaces on ordinal segments.

Extension of Gorak's previous classification results.

Abstract

We conclude the classification of spaces of continuous functions on ordinals carried out by R. Gorak. This gives a complete topological classification of the spaces $C_p([0,\alpha])$ of all continuous real-valued functions on compact segments of ordinals endowed with the topology of pointwise convergence. Moreover, this topological classification of the spaces $C_p([0,\alpha])$ completely coincides with their uniform classification.