Real Analysis Exchange

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

Leonard Huang

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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

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

First available in Project Euclid: 27 June 2018

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 03E10: Ordinal and cardinal numbers 54A05: Topological spaces and generalizations (closure spaces, etc.)
Secondary: 26A24: Differentiation (functions of one variable): general theory, generalized derivatives, mean-value theorems [See also 28A15]

cardinal ordinal order-embedding symmetric derivative Quasi-Mean Value Theorem


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.

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