First uncountable ordinal

In mathematics, the first uncountable ordinal, traditionally denoted by $\omega _{1}$ or sometimes by $\Omega$ , is the smallest ordinal number that, considered as a set, is uncountable. It is the supremum (least upper bound) of all countable ordinals. When considered as a set, the elements of $\omega _{1}$ are the countable ordinals (including finite ordinals),^[1] of which there are uncountably many.

Like any ordinal number (in von Neumann's approach), $\omega _{1}$ is a well-ordered set, with set membership serving as the order relation. $\omega _{1}$ is a limit ordinal, i.e. there is no ordinal $\alpha$ such that $\omega _{1}=\alpha +1$ .

The cardinality of the set $\omega _{1}$ is the first uncountable cardinal number, $\aleph _{1}$ (aleph-one). The ordinal $\omega _{1}$ is thus the initial ordinal of $\aleph _{1}$ . Under the continuum hypothesis, the cardinality of $\omega _{1}$ is $\beth _{1}$ , the same as that of $\mathbb {R}$ —the set of real numbers.^[2]

In most constructions, $\omega _{1}$ and $\aleph _{1}$ are considered equal as sets. To generalize: if $\alpha$ is an arbitrary ordinal, we define $\omega _{\alpha }$ as the initial ordinal of the cardinal $\aleph _{\alpha }$ .

The existence of $\omega _{1}$ 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, $\omega _{1}$ is often written as $[0,\omega _{1})$ , to emphasize that it is the space consisting of all ordinals smaller than $\omega _{1}$ .

If the axiom of countable choice holds, every increasing ω-sequence of elements of $[0,\omega _{1})$ converges to a limit in $[0,\omega _{1})$ . The reason is that the union (i.e., supremum) of every countable set of countable ordinals is another countable ordinal.

The topological space $[0,\omega _{1})$ is sequentially compact, but not compact. As a consequence, it is not metrizable. It is, however, countably compact and thus not Lindelöf (a countably compact space is compact if and only if it is Lindelöf). In terms of axioms of countability, $[0,\omega _{1})$ is first-countable, but neither separable nor second-countable.

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

References

^ "Set Theory > Basic Set Theory (Stanford Encyclopedia of Philosophy)". plato.stanford.edu. Retrieved 2020-08-12.
^ "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).