First uncountable ordinal

Template:Short description In mathematics, the first uncountable ordinal, traditionally denoted by ${\omega }_1$ or sometimes by $\mathrm{\Omega }$, is the smallest ordinal number that is the order type of an uncountable well-ordered set. It is the supremum (least upper bound) of all countable ordinals. In the von Neumann representation, the elements of ${\omega }_1$ are the countable ordinals (including finite ordinals), of which there are uncountably many.

The cardinality of the set ${\omega }_1$ is the first uncountable cardinal number, ${\mathrm{\aleph }}_1$ (aleph-one). The ordinal ${\omega }_1$ is thus the initial ordinal of Template:Tmath. Like all other initial ordinals of infinite cardinals, ${\omega }_1$ is a limit ordinal, i.e. there is no ordinal $\alpha$ such that Template:Tmath. Formally, cardinal numbers are usually represented as their initial ordinals, in which case ${\omega }_1$ and ${\mathrm{\aleph }}_1$ are considered equal as sets. More generally, for any ordinal Template:Tmath, ${\omega }_{\alpha }$ denotes the initial ordinal of the cardinal Template:Tmath.

The continuum hypothesis (CH) states that Template:Tmath (where Template:Tmath is the second beth number), which implies that Template:Tmath, i.e., the countable ordinals are equinumerous to the real numbers. If CH does not hold, but the axiom of choice (AC) does, then Template:Tmath, as the smallest uncountable cardinal, is strictly less than Template:Tmath. If AC also does not hold then Template:Tmath may be incomparable with Template:Tmath, but never larger than Template:Tmath.

The existence of ${\omega }_1$ does not depend on AC, as it can be constructed explicitly as the Hartogs number of Template:Tmath. More concretely, the set of all well-orderings on Template:Tmath can be constructed as a subset of all binary relations on Template:Tmath, and thus applying the axiom of replacement to replace every well-ordering with its order type will give Template:Tmath.

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.

See also

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

Bibliography

• Thomas Jech, Set Theory, 3rd millennium ed., 2003, Springer Monographs in Mathematics, Springer, Template:ISBN.
• Lynn Arthur Steen and J. Arthur Seebach, Jr., Counterexamples in Topology. Springer-Verlag, New York, 1978. Reprinted by Dover Publications, New York, 1995. Template:ISBN (Dover edition).