First uncountable ordinal

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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 ωα as the initial ordinal of the cardinal ℵα.

The existence of ω1 can be proven without the axiom of choice. (See Hartogs number.)

Topological properties[edit]

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[edit]