First uncountable ordinal
In mathematics, the first uncountable ordinal, traditionally denoted by [math]\displaystyle{ \omega_1 }[/math] or sometimes by [math]\displaystyle{ \Omega }[/math], 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 [math]\displaystyle{ \omega_1 }[/math] are the countable ordinals (including finite ordinals),[1] of which there are uncountably many.
Like any ordinal number (in von Neumann's approach), [math]\displaystyle{ \omega_1 }[/math] is a well-ordered set, with set membership serving as the order relation. [math]\displaystyle{ \omega_1 }[/math] is a limit ordinal, i.e. there is no ordinal [math]\displaystyle{ \alpha }[/math] such that [math]\displaystyle{ \omega_1 = \alpha+1 }[/math].
The cardinality of the set [math]\displaystyle{ \omega_1 }[/math] is the first uncountable cardinal number, [math]\displaystyle{ \aleph_1 }[/math] (aleph-one). The ordinal [math]\displaystyle{ \omega_1 }[/math] is thus the initial ordinal of [math]\displaystyle{ \aleph_1 }[/math]. Under the continuum hypothesis, the cardinality of [math]\displaystyle{ \omega_1 }[/math] is [math]\displaystyle{ \beth_1 }[/math], the same as that of [math]\displaystyle{ \mathbb{R} }[/math]—the set of real numbers.[2]
In most constructions, [math]\displaystyle{ \omega_1 }[/math] and [math]\displaystyle{ \aleph_1 }[/math] are considered equal as sets. To generalize: if [math]\displaystyle{ \alpha }[/math] is an arbitrary ordinal, we define [math]\displaystyle{ \omega_\alpha }[/math] as the initial ordinal of the cardinal [math]\displaystyle{ \aleph_\alpha }[/math].
The existence of [math]\displaystyle{ \omega_1 }[/math] 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, [math]\displaystyle{ \omega_1 }[/math] is often written as [math]\displaystyle{ [0,\omega_1) }[/math], to emphasize that it is the space consisting of all ordinals smaller than [math]\displaystyle{ \omega_1 }[/math].
If the axiom of countable choice holds, every increasing ω-sequence of elements of [math]\displaystyle{ [0,\omega_1) }[/math] converges to a limit in [math]\displaystyle{ [0,\omega_1) }[/math]. The reason is that the union (i.e., supremum) of every countable set of countable ordinals is another countable ordinal.
The topological space [math]\displaystyle{ [0,\omega_1) }[/math] 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, [math]\displaystyle{ [0,\omega_1) }[/math] is first-countable, but neither separable nor second-countable.
The space [math]\displaystyle{ [0,\omega_1]=\omega_1 + 1 }[/math] is compact and not first-countable. [math]\displaystyle{ \omega_1 }[/math] is used to define the long line and the Tychonoff plank—two important counterexamples in topology.
See also
References
Bibliography
- Thomas Jech, Set Theory, 3rd millennium ed., 2003, Springer Monographs in Mathematics, Springer, ISBN:3-540-44085-2.
- Lynn Arthur Steen and J. Arthur Seebach, Jr., Counterexamples in Topology. Springer-Verlag, New York, 1978. Reprinted by Dover Publications, New York, 1995. ISBN:0-486-68735-X (Dover edition).
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