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
or sometimes by
, 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
are the countable ordinals (including finite ordinals), of which there are uncountably many.
Like any ordinal number (in von Neumann's approach),
is a
well-ordered set, with
set membership serving as the order relation.
is a
limit ordinal, i.e. there is no ordinal
such that
.
The
cardinality of the set
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-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
is thus the
initial ordinal of
. Under the
continuum hypothesis, the cardinality of
is
, the same as that of
—the set of
real numbers.
In most constructions,
and
are considered equal as sets. To generalize: if
is an arbitrary ordinal, we define
as the initial ordinal of the cardinal
.
The existence of
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,
is often written as
,_to_emphasize_that_it_is_the_space_consisting_of_all_ordinals_smaller_than_
\omega_1.
If_the_axiom_of_countable_choice_holds,_every_
,_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_
,_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).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 ...
. 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
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
is first-countable space">first-countable, but neither separable space">separable nor second-countable space">second-countable.
The space
is compact and not first-countable.
is used to define the long line (topology), long line and the Tychonoff plank—two important counterexamples in
.
* Thomas Jech, ''Set Theory'', 3rd millennium ed., 2003, Springer Monographs in Mathematics, Springer, .
* Lynn Arthur Steen and J. Arthur Seebach, Jr., ''
''. Springer-Verlag, New York, 1978. Reprinted by Dover Publications, New York, 1995. {{ISBN, 0-486-68735-X (Dover edition).