In mathematics, the first uncountable ordinal, traditionally denoted by
, 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
are the countable ordinals (including finite ordinals),[1] of which there are uncountably many.
Like any ordinal number (in von Neumann's approach),
is a well-ordered set, with set membership serving as the order relation.
The cardinality of the set
is the first uncountable cardinal number,
is thus the initial ordinal of
Under the continuum hypothesis, the cardinality of
—the set of real numbers.
are considered equal as sets.
is an arbitrary ordinal, we define
as the initial ordinal of the cardinal
can be proven without the axiom of choice.
Any ordinal number can be turned into a topological space by using the order topology.
When viewed as a topological space,
, to emphasize that it is the space consisting of all ordinals smaller than
If the axiom of countable choice holds, every increasing ω-sequence of elements of
converges to a limit in
The reason is that the union (i.e., supremum) of every countable set of countable ordinals is another countable ordinal.
The topological space
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,
is first-countable, but neither separable nor second-countable.
is compact and not first-countable.
is used to define the long line and the Tychonoff plank—two important counterexamples in topology.