First 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">axiom of countable choice holds, every increasing ω-sequence of elements of limit in [0,\omega_1">sequence">increasing ω-sequence of elements of [0,\omega_1/math> converges to a Limit of a sequence">limit in [0,\omega_1/math>. The reason is that the union (set theory)">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 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, first-countable, but neither separable space">separable nor second-countable space">second-countable