First uncountable ordinal

In mathematics, the first uncountable ordinal, traditionally denoted by ω₁ or sometimes by Ω,[1] is the smallest ordinal number that, considered as a set, is uncountable. It is the supremum (least upper bound) of all countable ordinals. The elements of ω₁ are the countable ordinals (including finite ordinals),[2] of which there are uncountably many.

Like any ordinal number (in von Neumann's approach), ω₁ is a well-ordered set, with set membership ("∈") serving as the order relation. ω₁ is a limit ordinal, i.e. there is no ordinal α with α + 1 = ω₁.

The cardinality of the set ω₁ is the first uncountable cardinal number, ℵ₁ (aleph-one). The ordinal ω₁ is thus the initial ordinal of ℵ₁. Under continuum hypothesis, the cardinality of ω₁ is the same as that of ${\displaystyle \mathbb {R} }$—the set of real numbers.[3]

In most constructions, ω₁ and ℵ₁ are considered equal as sets. To generalize: if α is an arbitrary ordinal, we define ω_α as the initial ordinal of the cardinal ℵ_α.

The existence of ω₁ can be proven without the axiom of choice. For more, see Hartogs number.

Topological properties

Any ordinal number can be turned into a topological space by using the order topology. When viewed as a topological space, ω₁ is often written as [0,ω₁), to emphasize that it is the space consisting of all ordinals smaller than ω₁.

If the axiom of countable choice holds, every increasing ω-sequence of elements of [0,ω₁) converges to a limit in [0,ω₁). The reason is that the union (i.e., supremum) of every countable set of countable ordinals is another countable ordinal.

The topological space [0,ω₁) is sequentially compact, but not compact. As a consequence, it is not metrizable. It is, however, countably compact and thus not Lindelöf. In terms of axioms of countability, [0,ω₁) is first-countable, but neither separable nor second-countable.

The space [0, ω₁] = ω₁ + 1 is compact and not first-countable. ω₁ is used to define the long line and the Tychonoff plank—two important counterexamples in topology.

See also

• Epsilon numbers (mathematics)
• Large countable ordinal
• Ordinal arithmetic

References

1. "Comprehensive List of Set Theory Symbols". Math Vault. 2020-04-11. Retrieved 2020-08-12.
2. "Set Theory > Basic Set Theory (Stanford Encyclopedia of Philosophy)". plato.stanford.edu. Retrieved 2020-08-12.
3. "first uncountable ordinal in nLab". ncatlab.org. Retrieved 2020-08-12.

Bibliography

• Thomas Jech, Set Theory, 3rd millennium ed., 2003, Springer Monographs in Mathematics, Springer, ISBN 3-540-44085-2.
• Lynn Arthur Steen and J. Arthur Seebach, Jr., Counterexamples in Topology. Springer-Verlag, New York, 1978. Reprinted by Dover Publications, New York, 1995. ISBN 0-486-68735-X (Dover edition).