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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 and their changes (cal ...
, the first uncountable ordinal, traditionally denoted by ω1 or sometimes by Ω, is the smallest
ordinal 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 ...

uncountable
. 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 ...

supremum
(least upper bound) of all countable ordinals. The elements of ω1 are the countable ordinals (including finite ordinals), of which there are uncountably many. Like any ordinal number (in von Neumann's approach), ω1 is a
well-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. ω1 is a
limit 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 = ω1. The
cardinality 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 ω1 is the first uncountable
cardinal 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 ...
, ℵ1 ( aleph-one). The ordinal ω1 is thus the
initial 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 ℵ1. Under the
continuum 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 ω1 is the same as that of \mathbb—the set of
real 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, ω1 and ℵ1 are considered equal as sets. To generalize: if α is an arbitrary ordinal, we define ωα as the initial ordinal of the cardinal ℵα. The existence of ω1 can be proven without the
axiom 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 ...

axiom of choice
. 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 a
topological 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, ω1 is often written as 1),_to_emphasize_that_it_is_the_space_consisting_of_all_ordinals_smaller_than_ω1. If_the_axiom_of_countable_choice_holds,_every_
1),_to_emphasize_that_it_is_the_space_consisting_of_all_ordinals_smaller_than_ω1. If_the_axiom_of_countable_choice_holds,_every_sequence">increasing_ω-sequence_of_elements_of_ 1),_to_emphasize_that_it_is_the_space_consisting_of_all_ordinals_smaller_than_ω1. If_the_axiom_of_countable_choice_holds,_every_sequence">increasing_ω-sequence_of_elements_of_[0,ω1)_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)._The_reason_is_that_the_union_(set_theory).html" ;"title=",ω1)_converges_to_a_Limit_of_a_sequence.html" "title="sequence.html" ;"title="axiom_of_countable_choice.html" ;"title=",ω1), to emphasize that it is the space consisting of all ordinals smaller than ω1. If the axiom of countable choice">,ω1), to emphasize that it is the space consisting of all ordinals smaller than ω1. If the axiom of countable choice holds, every sequence">increasing ω-sequence of elements of [0,ω1) 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). 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,ω1) is sequentially compact, but not compact space, compact. As a consequence, it is not metrizable space, metrizable. It is, however, countably compact space, countably compact and thus not Lindelöf space, Lindelöf. In terms of axioms of countability,
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_
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=",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 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, ω1] = ω1 + 1 is compact and not first-countable. ω1 is used to define the long line (topology), long line and the Tychonoff plank—two important counterexamples in
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 ...

topology
.


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