# First uncountable ordinal

In mathematics, the **first uncountable ordinal**, traditionally denoted by **ω _{1}** or sometimes by

**Ω**,

^{[1]}is the smallest ordinal number that, considered as a set, is uncountable. It is the supremum (least upper bound) of all countable ordinals. The elements of ω

_{1}are the countable ordinals (including finite ordinals),

^{[2]}of which there are uncountably many.

Like any ordinal number (in von Neumann's approach), ω_{1} is a well-ordered set, with set membership ("∈") serving as the order relation. ω_{1} is a limit ordinal, i.e. there is no ordinal α with α + 1 = ω_{1}.

The cardinality of the set ω_{1} is the first uncountable cardinal number, ℵ_{1} (aleph-one). The ordinal ω_{1} is thus the initial ordinal of ℵ_{1}. Under continuum hypothesis, the cardinality of ω_{1} is the same as that of —the set of real numbers.^{[3]}

In most constructions, ω_{1} and ℵ_{1} are considered equal as sets. To generalize: if α is an arbitrary ordinal, we define ω_{α} as the initial ordinal of the cardinal ℵ_{α}.

The existence of ω_{1} can be proven without the axiom of choice. For more, see Hartogs number.

## Topological propertiesEdit

Any ordinal number can be turned into a topological space by using the order topology. When viewed as a topological space, ω_{1} is often written as [0,ω_{1}), to emphasize that it is the space consisting of all ordinals smaller than ω_{1}.

If the axiom of countable choice holds, every increasing ω-sequence of elements of [0,ω_{1}) converges to a limit in [0,ω_{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,ω_{1}) is sequentially compact, but not compact. As a consequence, it is not metrizable. It is, however, countably compact and thus not Lindelöf. In terms of axioms of countability, [0,ω_{1}) is first-countable, but neither separable nor second-countable.

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

## See alsoEdit

## ReferencesEdit

**^**"Comprehensive List of Set Theory Symbols".*Math Vault*. 2020-04-11. Retrieved 2020-08-12.**^**"Set Theory > Basic Set Theory (Stanford Encyclopedia of Philosophy)".*plato.stanford.edu*. Retrieved 2020-08-12.**^**"first uncountable ordinal in nLab".*ncatlab.org*. Retrieved 2020-08-12.

## BibliographyEdit

- 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).