First Uncountable Ordinal

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## Topological properties

## See also

## References

## Bibliography

This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

First Uncountable Ordinal

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

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

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

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

The existence of ?_{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, ?_{1} is often written as [0,?_{1}), to emphasize that it is the space consisting of all ordinals smaller than ?_{1}.

If the axiom of countable choice holds, every increasing ω-sequence of elements of [0,?_{1}) converges to a limit in [0,?_{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,?_{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,?_{1}) is first-countable, but neither separable nor second-countable.

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

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

- 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).

This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

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