Some Applications of Order-Embeddings of Countable Ordinals into the Real Line

Abstract

It is a well-known fact that an ordinal \( \alpha \) can be embedded into the real line \( \mathbb{R} \) in an order-preserving manner if and only if \( \alpha \) is countable. However, it would seem that outside of set theory, this fact has not yet found any concrete applications. The goal of this paper is to present some applications. More precisely, we show how two classical results, one in point-set topology and the other in real analysis, can be proven by defining specific order-embeddings of countable ordinals into \( \mathbb{R} \).