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 ωα as the initial ordinal of the cardinal ℵα.

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.