In _{1} or sometimes by Ω, is the smallest _{1} are the countable ordinals (including finite ordinals), of which there are uncountably many.
Like any ordinal number (in von Neumann's approach), ω_{1} is a _{1} is a _{1}.
The _{1} is the first uncountable _{1} ( aleph-one). The ordinal ω_{1} is thus the _{1}. Under the _{1} is the same as that of $\backslash mathbb$—the set of _{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

_{1} is often written as 1),_to_emphasize_that_it_is_the_space_consisting_of_all_ordinals_smaller_than_ω1.
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If the axiom of countable choice">,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 sequence">increasing ω-sequence of elements of [0,ω_{1}) converges to a Limit of a sequence">limit
Limit or Limits may refer to:
Arts and media
* Limit (music), a way to characterize harmony
* Limit (song), "Limit" (song), a 2016 single by Luna Sea
* Limits (Paenda song), "Limits" (Paenda song), 2019 song that represented Austria in the Eurov ...

in [0,ωmathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...

, the first uncountable ordinal, traditionally denoted by ωordinal number
In set theory
Set theory is the branch of that studies , which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of , is mostly concerned with those that ...

that, considered as a set, is uncountable
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

. It is the supremum
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...

(least upper bound) of all countable ordinals. The elements of ωwell-ordered set
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

, with set membership
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

("∈") serving as the order relation. ωlimit ordinal
In set theory, a limit ordinal is an ordinal number that is neither zero nor a successor ordinal. Alternatively, an ordinal λ is a limit ordinal if there is an ordinal less than λ, and whenever β is an ordinal less than λ, then there exists an ...

, i.e. there is no ordinal α with α + 1 = ωcardinality
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

of the set ωcardinal number
150px, Aleph null, the smallest infinite cardinal
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and ca ...

, ℵinitial ordinal
The von Neumann cardinal assignment is a cardinal assignment which uses ordinal number
In set theory, an ordinal number, or ordinal, is one generalization of the concept of a natural number that is used to describe a way to arrange a (possibly ...

of ℵcontinuum hypothesis
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

, the cardinality of ωreal numbers
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

.
In most constructions, ωaxiom of choice
In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product#Infinite Cartesian products, Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the a ...

. For more, see Hartogs number
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

.
Topological properties

Any ordinal number can be turned into atopological space
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...

by using the order topology
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

. When viewed as a topological space, ω,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_sequence">increasing_ω-sequence_of_elements_of_ ,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_sequence">increasing_ω-sequence_of_elements_of_[0,ω_{1})_converges_to_a_Limit_of_a_sequence">limit
Limit_or_Limits_may_refer_to:
_Arts_and_media
*_Limit_(music),_a_way_to_characterize_harmony
*_Limit_(song),_"Limit"_(song),_a_2016_single_by_Luna_Sea
*_Limits_(Paenda_song),_"Limits"_(Paenda_song),_2019_song_that_represented_Austria_in_the_Eurov_...

_in_[0,ω1)_is_first-countable_space">first-countable
In_topology
s,_which_have_only_one_surface_and_one_edge,_are_a_kind_of_object_studied_in_topology.
In_mathematics,_topology_(from_the_Greek_language,_Greek_words_,_and_)_is_concerned_with_the_properties_of_a_mathematical_object,_geometric_objec_...

,_but_neither_1)_is_first-countable_space">first-countable
In_topology
s,_which_have_only_one_surface_and_one_edge,_are_a_kind_of_object_studied_in_topology.
In_mathematics,_topology_(from_the_Greek_language,_Greek_words_,_and_)_is_concerned_with_the_properties_of_a_mathematical_object,_geometric_objec_...

,_but_neither_separable_space">separable_nor_second-countable_space.html" "title="separable_space.html" ;"title="first-countable_space.html" "title=",ω, but neither separable space">separable nor second-countable space">second-countable In topology, a second-countable space, also called a completely separable space, is a topological space whose topology has a countable base (topology), base. More explicitly, a topological space T is second-countable if there exists some countable ...

. The space [0, ω

topology
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

.
See also

*Epsilon numbers (mathematics)
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

* Large countable ordinal
In the mathematical discipline of set theory, there are many ways of describing specific countable set, countable ordinal number, ordinals. The smallest ones can be usefully and non-circularly expressed in terms of their Cantor normal forms. Beyond ...

* Ordinal arithmetic
In the mathematical field of set theory
illustrating the intersection (set theory), intersection of two set (mathematics), sets.
Set theory is a branch of mathematical logic that studies Set (mathematics), sets, which informally are collections ...

References

Bibliography

* Thomas Jech, ''Set Theory'', 3rd millennium ed., 2003, Springer Monographs in Mathematics, Springer, . * Lynn Arthur Steen and J. Arthur Seebach, Jr., ''Counterexamples in Topology
''Counterexamples in Topology'' (1970, 2nd ed. 1978) is a book on mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geomet ...

''. Springer-Verlag, New York, 1978. Reprinted by Dover Publications, New York, 1995. {{ISBN, 0-486-68735-X (Dover edition).
Ordinal numbers
Topological spaces