First uncountable ordinal

{{Short description|Smallest ordinal number that, considered as a set, is uncountable}}

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),{{Cite web|title=Set Theory > Basic Set Theory (Stanford Encyclopedia of Philosophy)|url=https://plato.stanford.edu/entries/set-theory/basic-set-theory.html|access-date=2020-08-12|website=plato.stanford.edu}} 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.{{Cite web|title=first uncountable ordinal in nLab|url=https://ncatlab.org/nlab/show/first+uncountable+ordinal|access-date=2020-08-12|website=ncatlab.org}}

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

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).

Category:Ordinal numbers

Category:Topological spaces

First uncountable ordinal

Topological properties

See also

References

Bibliography