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.

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 $\aleph _{\alpha }$.

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 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.

^"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.

Bibliographyedit

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).