Smallest 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), 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 $\mathbb$—the set of real numbers.

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 ,\omega_1), to emphasize that it is the space consisting of all ordinals smaller than $\omega_1$.

If the increasing ω-sequence of elements of $[0,\omega_1)$ converges to a Limit of a sequence">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 union (set theory)">union (i.e., supremum) of every countable set of countable ordinals is another countable ordinal.

The topological space compact. As a consequence, it is not metrizable space">metrizable. It is, however, countably compact space">countably compact and thus not Lindelöf space">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 space, first-countable, but neither separable space, separable nor second-countable space, 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.

See also

* Epsilon numbers (mathematics)
* Large countable ordinal
* Ordinal arithmetic

References

Bibliography

* Thomas Jech, ''Set Theory'', 3rd millennium ed., 2003, Springer Monographs in Mathematics, Springer, .
* 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).

Ordinal numbers
Topological spaces