First uncountable ordinal

In mathematics, the first uncountable ordinal, traditionally denoted by ${\displaystyle \omega _{1}}$ (or sometimes ${\displaystyle \Omega }$), is the smallest ordinal number which, when viewed as a set, is uncountable (i.e. it does not have the same cardinality as the set ${\displaystyle \mathbb {N} }$ of natural numbers). Equivalently, ${\displaystyle \omega _{1}}$ 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 ${\displaystyle \in }$, and ${\displaystyle \alpha <\beta }$ iff ${\displaystyle \alpha \in \beta }$. Thus, when considered as a set, the elements of ${\displaystyle \omega _{1}}$ are precisely the countable ordinals (including the finite ordinals ${\displaystyle 0,1,2,\dots }$), of which there are uncountably many.^[1]

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

The cardinality of the set ${\displaystyle \omega _{1}}$ is the first uncountable cardinal, denoted ${\displaystyle \aleph _{1}}$ (aleph-one). The ordinal ${\displaystyle \omega _{1}}$ is therefore the initial ordinal of the cardinal ${\displaystyle \aleph _{1}}$ (an initial ordinal is the least ordinal of a given cardinality). It is common in set theory to identify each infinite cardinal ${\displaystyle \aleph _{\alpha }}$ with its initial ordinal ${\displaystyle \omega _{\alpha }}$, so that as sets one may write ${\displaystyle \omega _{\alpha }=\aleph _{\alpha }}$. More generally, for any ordinal ${\displaystyle \alpha }$, ${\displaystyle \omega _{\alpha }}$ denotes the initial ordinal of the cardinal ${\displaystyle \aleph _{\alpha }}$.

Under the continuum hypothesis (CH)—the statement that there is no set whose cardinality lies strictly between that of ${\displaystyle \mathbb {N} }$ and that of ${\displaystyle \mathbb {R} }$—one has ${\displaystyle |\mathbb {R} |=\aleph _{1}}$. In that case the cardinality of ${\displaystyle \omega _{1}}$ is also ${\displaystyle \beth _{1}}$ (the second beth number), the same cardinality as the set ${\displaystyle \mathbb {R} }$ of real numbers.^[2]

The existence of ${\displaystyle \omega _{1}}$ does not require the full axiom of choice (AC). Indeed, for any set ${\displaystyle X}$, the Hartogs number ${\displaystyle \aleph (X)}$ is the least ordinal that cannot be injected into ${\displaystyle X}$; taking ${\displaystyle X=\mathbb {N} }$ yields an uncountable ordinal, which (by definition) is at least as large as ${\displaystyle \omega _{1}}$. In particular, ${\displaystyle \omega _{1}}$ exists in ZF without AC. (Here, a set is countable if it is finite or countably infinite, i.e., in bijection with ${\displaystyle \mathbb {N} }$; otherwise it is uncountable.)

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

Any ordinal can be viewed as a topological space by giving it the order topology: a base is formed by open intervals ${\displaystyle (\alpha ,\beta )}$ together with initial segments of the form ${\displaystyle [0,\beta )}$ and, when the top element is present, final segments of the form ${\displaystyle (\alpha ,\gamma ]}$. When considered with this topology, the space is denoted ${\displaystyle [0,\gamma )}$ or ${\displaystyle [0,\gamma ]}$ as above.

If the axiom of countable choice (CC) holds, every increasing ${\displaystyle \omega }$-sequence (i.e., a sequence indexed by the natural numbers) in ${\displaystyle [0,\omega _{1}]}$ 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 ${\displaystyle \langle \alpha _{n}:n\in \omega \rangle }$ has limit ${\displaystyle \sup _{n}\alpha _{n}\leq \omega _{1}}$, which lies in ${\displaystyle [0,\omega _{1}]}$.

The space ${\displaystyle [0,\omega _{1})}$ 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, ${\displaystyle [0,\omega _{1})}$ 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 ${\displaystyle [0,\omega _{1})}$ is not Lindelöf. In terms of axioms of countability, ${\displaystyle [0,\omega _{1})}$ 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 ${\displaystyle [0,\omega _{1}]=\omega _{1}+1}$ is compact (every open cover has a finite subcover) but not first-countable: the top point ${\displaystyle \omega _{1}}$ has cofinality ${\displaystyle \omega _{1}}$ (uncountable), so no countable neighborhood base can converge to it in the order topology.

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

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