- You are here
- Everything Explained.Today
- A-Z Contents
- F
- FI
- FIR
- First uncountable ordinal

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

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

R

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.

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<sub>1</sub>), to emphasize that it is the space consisting of all ordinals smaller than ω<sub>1</sub>.
If the [[axiom of countable choice]] holds, every increasing ω-sequence of elements of [0,ω<sub>1</sub>) converges to a [[Limit of a sequence|limit]] in [0,ω<sub>1</sub>). The reason is that the [[union (set theory)|union]] (i.e., supremum) of every countable set of countable ordinals is another countable ordinal.

The topological space [0,ω<sub>1</sub>) 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<sub>1</sub>) is [[first-countable space|first-countable]], but neither separable nor second-countable.

The space [0, ω<sub>1</sub>] = 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.

- Web site: 2020-04-11. Comprehensive List of Set Theory Symbols. 2020-08-12. Math Vault. en-US.
- Web site: Set Theory > Basic Set Theory (Stanford Encyclopedia of Philosophy). 2020-08-12. plato.stanford.edu.
- Web site: first uncountable ordinal in nLab. 2020-08-12. ncatlab.org.

- Thomas Jech,
*Set Theory*, 3rd millennium ed., 2003, Springer Monographs in Mathematics, Springer, . - Lynn Arthur Steen and J. Arthur Seebach, Jr.,
*Counterexamples in Topology*. Springer-Verlag, New York, 1978. Reprinted by Dover Publications, New York, 1995. (Dover edition).