## Real Analysis Exchange

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

Leonard Huang

#### 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 $\mathbb{R}$.

#### Article information

Source
Real Anal. Exchange, Volume 43, Number 2 (2018), 417-428.

Dates
First available in Project Euclid: 27 June 2018

Permanent link to this document
https://projecteuclid.org/euclid.rae/1530064970

Digital Object Identifier
doi:10.14321/realanalexch.43.2.0417

Mathematical Reviews number (MathSciNet)
MR3942587

Zentralblatt MATH identifier
06924898

#### Citation

Huang, Leonard. Some Applications of Order-Embeddings of Countable Ordinals into the Real Line. Real Anal. Exchange 43 (2018), no. 2, 417--428. doi:10.14321/realanalexch.43.2.0417. https://projecteuclid.org/euclid.rae/1530064970