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 $$\mathbb{R}$$—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 $$[0,\omega_1)$$, to emphasize that it is the space consisting of all ordinals smaller than $$\omega_1$$.

If the axiom of countable choice holds, every increasing &omega;-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.