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 $\mathrm{\Omega }$), is the smallest ordinal number that, when viewed as a set, is uncountable (i.e. it does not have the same cardinality as a subset of the set $\mathbb{\mathbb{N}}$ of natural numbers). Equivalently, ${\omega }_1$ is the supremum (least upper bound) of all countable ordinals. In the standard von Neumann ordinal approach, an ordinal is a transitive set well-ordered by the membership relation $\in$, and $\alpha <\beta$ iff $\alpha \in \beta$. Thus, when considered as a set, the elements of ${\omega }_1$ are precisely the countable ordinals (including the finite ordinals $0,1,2,\dots$), of which there are uncountably many.

Like any ordinal number, ${\omega }_1$ is a well-ordered set. It is a limit ordinal (an ordinal with no immediate predecessor): there is no ordinal $\alpha$ such that ${\omega }_1=\alpha +1$.

The cardinality of the set ${\omega }_1$ is the first uncountable cardinal, denoted ${\mathrm{\aleph }}_1$ (aleph-one). The ordinal ${\omega }_1$ is therefore the initial ordinal of the cardinal ${\mathrm{\aleph }}_1$ (an initial ordinal is the least ordinal of a given cardinality). It is common in set theory to identify each infinite cardinal ${\mathrm{\aleph }}_{\alpha }$ with its initial ordinal ${\omega }_{\alpha }$, so that as sets one may write ${\omega }_{\alpha }={\mathrm{\aleph }}_{\alpha }$. More generally, for any ordinal $\alpha$, ${\omega }_{\alpha }$ denotes the initial ordinal of the cardinal ${\mathrm{\aleph }}_{\alpha }$.

Under the continuum hypothesis (CH)—the statement that there is no set whose cardinality lies strictly between that of $\mathbb{\mathbb{N}}$ and that of $\mathbb{ℝ}$—one has $|\mathbb{ℝ}|={\mathrm{\aleph }}_1$. In that case the cardinality of ${\omega }_1$ is also ${\beth }_1$ (the second beth number), the same cardinality as the set $\mathbb{ℝ}$ of real numbers.

The existence of ${\omega }_1$ does not require the full axiom of choice (AC). Indeed, for any set $X$, the Hartogs number $\mathrm{\aleph }(X)$ is the least ordinal that cannot be injected into $X$; taking $X=\mathbb{\mathbb{N}}$ yields an uncountable ordinal, which (by definition) is at least as large as ${\omega }_1$. In particular, ${\omega }_1$ exists in ZF without AC. (Here, a set is countable if it is finite or countably infinite, i.e., in bijection with $\mathbb{\mathbb{N}}$; otherwise it is uncountable.)

Topological properties

For ordinal intervals, we write $[0,\gamma )$ for the set of all ordinals $\alpha$ with $0\le \alpha <\gamma$, equipped with the order topology (see below). The space $[0,{\omega }_1)$ thus consists of all ordinals strictly less than ${\omega }_1$, while $[0,{\omega }_1]={\omega }_1+1$ includes the point ${\omega }_1$ as a top element.

Any ordinal gives rise to a topological space by equipping it with the order topology: a base is formed by open intervals $(\alpha ,\beta )$ together with initial segments of the form $[0,\beta )$ and, when the top element is present, final segments of the form $(\alpha ,\gamma ]$. When considered with this topology, the space is again denoted $[0,\gamma )$ or $[0,\gamma ]$ as above.

If the axiom of countable choice (CC) holds, every increasing $\omega$-sequence (i.e., a sequence indexed by the natural numbers) in $[0,{\omega }_1]$ converges. Indeed, the pointwise union (which is the supremum in the ordinal order) of a countable set of countable ordinals is again a countable ordinal; therefore any increasing sequence $⟨{\alpha }_n:n\in \omega ⟩$ has limit $\underset{n}{\sup }{\alpha }_n\le {\omega }_1$, which lies in $[0,{\omega }_1]$.

The space $[0,{\omega }_1)$ is sequentially compact (every sequence has a convergent subsequence) but not compact (there exist open covers with no finite subcover). Consequently, it is not metrizable (every compact metric space is sequentially compact and conversely, but a non-compact sequentially compact space cannot be metric). Nevertheless, $[0,{\omega }_1)$ is countably compact (every countable open cover admits a finite subcover; equivalently, every countably infinite subset has a limit point); since a space is compact iff it is both countably compact and Lindelöf (every open cover has a countable subcover), it follows that $[0,{\omega }_1)$ is not Lindelöf. In terms of axioms of countability, $[0,{\omega }_1)$ is first-countable (every point has a countable local base), but it is neither separable (it has no countable dense subset) nor second-countable (it has no countable base).

By contrast, the space $[0,{\omega }_1]={\omega }_1+1$ is compact (every open cover has a finite subcover) but not first-countable: the top point ${\omega }_1$ has cofinality ${\omega }_1$ (uncountable), so no countable neighborhood base can converge to it in the order topology.

The ordinal ${\omega }_1$ is a standard building block for classical counterexamples in topology. The long line is obtained by taking the lexicographic order on ${\omega }_1\times [0,1)$ and forming the associated order topology; it is locally like $\mathbb{ℝ}$ but not second-countable and not paracompact. The Tychonoff plank is the product space $[0,{\omega }_1]\times [0,\omega ]$ (with the product of order topologies), which exhibits further separability and compactness pathologies.

See also

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

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).