First uncountable ordinal

The cardinality of the set ${\displaystyle {\omega }_1}$ is the first uncountable cardinal number, ${\displaystyle {\mathrm{\aleph }}_1}$ (aleph-one). The ordinal ${\displaystyle {\omega }_1}$ is thus the initial ordinal of ⁠${\displaystyle {\mathrm{\aleph }}_1}$⁠. Like all other initial ordinals of infinite cardinals, ${\displaystyle {\omega }_1}$ is a limit ordinal, i.e. there is no ordinal ${\displaystyle \alpha }$ such that ⁠${\displaystyle {\omega }_1=\alpha +1}$⁠. Formally, cardinal numbers are usually represented as their initial ordinals, in which case ${\displaystyle {\omega }_1}$ and ${\displaystyle {\mathrm{\aleph }}_1}$ are considered equal as sets. More generally, for any ordinal ⁠${\displaystyle \alpha }$⁠, ${\displaystyle {\omega }_{\alpha }}$ denotes the initial ordinal of the cardinal ⁠${\displaystyle {\mathrm{\aleph }}_{\alpha }}$⁠.

The continuum hypothesis (CH) states that ⁠${\displaystyle {\beth }_1={\mathrm{\aleph }}_1}$⁠ (where ⁠${\displaystyle {\beth }_1=2^{{\mathrm{\aleph }}_0}=|\mathbb{R}|}$⁠ is the second beth number), which implies that ⁠${\displaystyle |{\omega }_1|=|\mathbb{R}|}$⁠, i.e., the countable ordinals are equinumerous to the real numbers. If CH does not hold, but the axiom of choice (AC) does, then ⁠${\displaystyle |{\omega }_1|}$⁠, as the smallest uncountable cardinal, is strictly less than ⁠${\displaystyle |\mathbb{R}|}$⁠. If AC also does not hold then ⁠${\displaystyle |{\omega }_1|}$⁠ may be incomparable with ⁠${\displaystyle |\mathbb{R}|}$⁠, but never larger than ⁠${\displaystyle |\mathbb{R}|}$⁠.

The existence of ${\displaystyle {\omega }_1}$ does not depend on AC, as it can be constructed explicitly as the Hartogs number of ⁠${\displaystyle {\omega }_0=\mathbb{N}}$⁠. More concretely, the set of all well-orderings on ⁠${\displaystyle \mathbb{N}}$⁠ can be constructed as a subset of all binary relations on ⁠${\displaystyle \mathbb{N}}$⁠, and thus applying the axiom of replacement to replace every well-ordering with its order type will give ⁠${\displaystyle {\omega }_1}$⁠.

Topological properties

Any ordinal number can be turned into a topological space by using the order topology. When viewed as a topological space, ${\displaystyle {\omega }_1}$ is often written as ${\displaystyle [0,{\omega }_1)}$, to emphasize that it is the space consisting of all ordinals smaller than ${\displaystyle {\omega }_1}$.

If the axiom of countable choice holds, every increasing ω-sequence of elements of ${\displaystyle [0,{\omega }_1]}$ converges to a limit in ${\displaystyle [0,{\omega }_1]}$. The reason is that the union (i.e., supremum) of every countable set of countable ordinals is another countable ordinal.

The topological space ${\displaystyle [0,{\omega }_1)}$ 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, ${\displaystyle [0,{\omega }_1)}$ is first-countable, but neither separable nor second-countable.

The space ${\displaystyle [0,{\omega }_1]={\omega }_1+1}$ is compact and not first-countable. ${\displaystyle {\omega }_1}$ is used to define the long line and the Tychonoff plank—two important counterexamples in topology.