HOME

TheInfoList



OR:

In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the first uncountable ordinal, traditionally denoted by \omega_1 or sometimes by \Omega, is the smallest ordinal number that, considered as a
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
, is uncountable. It is the
supremum In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest ...
(least upper bound) of all countable ordinals. When considered as a set, the elements of \omega_1 are the countable ordinals (including finite ordinals), of which there are uncountably many. Like any ordinal number (in von Neumann's approach), \omega_1 is a well-ordered set, with set membership serving as the order relation. \omega_1 is a limit ordinal, i.e. there is no ordinal \alpha such that \omega_1 = \alpha+1. The cardinality of the set \omega_1 is the first uncountable
cardinal number In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets. The cardinality of a finite set is a natural number: the number of elements in the set. ...
, \aleph_1 (
aleph-one In mathematics, particularly in set theory, the aleph numbers are a sequence of numbers used to represent the cardinality (or size) of infinite sets that can be well-ordered. They were introduced by the mathematician Georg Cantor and are named a ...
). The ordinal \omega_1 is thus the initial ordinal of \aleph_1. Under the continuum hypothesis, the cardinality of \omega_1 is \beth_1, the same as that of \mathbb—the set of real numbers. In most constructions, \omega_1 and \aleph_1 are considered equal as sets. To generalize: if \alpha is an arbitrary ordinal, we define \omega_\alpha as the initial ordinal of the cardinal \aleph_\alpha. The existence of \omega_1 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 of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection ...
. For more, see Hartogs number.


Topological properties

Any ordinal number can be turned into a
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poin ...
by using the order topology. When viewed as a topological space, \omega_1 is often written as ,\omega_1),_to_emphasize_that_it_is_the_space_consisting_of_all_ordinals_smaller_than_\omega_1. If_the_axiom_of_countable_choice_holds,_every_
,\omega_1),_to_emphasize_that_it_is_the_space_consisting_of_all_ordinals_smaller_than_\omega_1. If_the_axiom_of_countable_choice_holds,_every_sequence">increasing_ω-sequence_of_elements_of_ ,\omega_1),_to_emphasize_that_it_is_the_space_consisting_of_all_ordinals_smaller_than_\omega_1. If_the_axiom_of_countable_choice_holds,_every_sequence">increasing_ω-sequence_of_elements_of_[0,\omega_1)_converges_to_a_Limit_of_a_sequence">limit_ Limit_or_Limits_may_refer_to: _Arts_and_media *__''Limit''_(manga),_a_manga_by_Keiko_Suenobu *__''Limit''_(film),_a_South_Korean_film *_Limit_(music),_a_way_to_characterize_harmony *__"Limit"_(song),_a_2016_single_by_Luna_Sea *_"Limits",_a_2019__...
_in_[0,\omega_1)._The_reason_is_that_the_union_(set_theory).html" ;"title=",\omega_1)_converges_to_a_Limit_of_a_sequence.html" "title="sequence.html" ;"title="axiom_of_countable_choice.html" ;"title=",\omega_1), to emphasize that it is the space consisting of all ordinals smaller than \omega_1. If the axiom of countable choice">,\omega_1), to emphasize that it is the space consisting of all ordinals smaller than \omega_1. If the axiom of countable choice holds, every sequence">increasing ω-sequence of elements of [0,\omega_1) converges to a Limit of a sequence">limit Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...
in [0,\omega_1). 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,\omega_1) 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 (a countably compact space is compact if and only if it is Lindelöf). In terms of axioms of countability, first-countable,_but_neither_ first-countable,_but_neither_separable_space">separable_nor_second-countable_space.html" ;"title="separable_space.html" ;"title="first-countable_space.html" ;"title=",\omega_1) is first-countable space">first-countable, but neither separable space">separable nor second-countable space">second-countable. The space [0,\omega_1]=\omega_1 + 1 is compact and not first-countable. \omega_1 is used to define the long line (topology), long line and the Tychonoff plank—two important counterexamples in
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
.


See also

* Epsilon numbers (mathematics) *
Large countable ordinal In the mathematical discipline of set theory, there are many ways of describing specific countable ordinals. The smallest ones can be usefully and non-circularly expressed in terms of their Cantor normal forms. Beyond that, many ordinals of relev ...
* Ordinal arithmetic


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 by topologists Lynn Steen and J. Arthur Seebach, Jr. In the process of working on problems like the metrization problem, topologists (including Steen and Seebach) h ...
''. Springer-Verlag, New York, 1978. Reprinted by Dover Publications, New York, 1995. {{ISBN, 0-486-68735-X (Dover edition). Ordinal numbers Topological spaces