Definition:Minimal Uncountable Well-Ordered Set
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Definition
Let $\Omega$ be an uncountable well-ordered set.
Then $\Omega$ is the minimal uncountable well-ordered set if and only if every initial segment in $\Omega$ is countable.
Also denoted as
This set is sometimes denoted $S_\Omega$, matching the notation of initial segments.
Also known as
The set $\Omega$ is also known as the set of countable ordinals.
Also see
- Existence of Minimal Uncountable Well-Ordered Set
- Set of Countable Ordinals Unique up to Isomorphism, justifying the use of "the" in the definition.
Sources
- 1984: Gerald B. Folland: Real Analysis: Modern Techniques and their Applications $\S \text P.18$
- 2000: James R. Munkres: Topology (2nd ed.) $\S 1.2$