First uncountable ordinal

In mathematics, the first uncountable ordinal, traditionally denoted by ω₁ or sometimes by Ω, is the smallest ordinal number that, considered as a set, is uncountable. It is the supremum of all countable ordinals. The elements of ω₁ are the countable ordinals, 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. Indeed, in most constructions ω₁ and ℵ₁ are 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. (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.

Every increasing ω-sequence of elements of [0,ω₁) converges to a limit in [0,ω₁). The reason is that the union (=supremum) of every countable set of countable ordinals is another countable ordinal.

The topological space [0,ω₁) is sequentially compact but not compact. It is however countably compact and thus not Lindelöf. In terms of axioms of countability, [0,ω₁) is first countable but not separable nor second countable. As a consequence, it is not metrizable.

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

• Ordinal arithmetic
• Large countable ordinal
• Continuum hypothesis

References

• 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).