In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the first uncountable ordinal, traditionally denoted by
or sometimes by
, is the smallest
ordinal number
In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, th, etc.) aimed to extend enumeration to infinite sets.
A finite set can be enumerated by successively labeling each element with the leas ...
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
In mathematics, an uncountable set, informally, is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal number is larger tha ...
. It is the
supremum
In mathematics, the infimum (abbreviated inf; : infima) of a subset S of a partially ordered set P is the greatest element in P that is less than or equal to each element of S, if such an element exists. If the infimum of S exists, it is unique, ...
(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
In mathematics, an element (or member) of a set is any one of the distinct objects that belong to that set. For example, given a set called containing the first four positive integers (A = \), one could say that "3 is an element of ", expressed ...
serving as the order relation.
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 a ...
, i.e. there is no ordinal
such that
.
The
cardinality
The thumb is the first digit of the hand, next to the index finger. When a person is standing in the medical anatomical position (where the palm is facing to the front), the thumb is the outermost digit. The Medical Latin English noun for thum ...
of the set
is the first uncountable
cardinal number
In mathematics, a cardinal number, or cardinal for short, is what is commonly called the number of elements of a set. In the case of a finite set, its cardinal number, or cardinality is therefore a natural number. For dealing with the cas ...
,
(
aleph-one). The ordinal
is thus the
initial ordinal of
. Under the
continuum hypothesis
In mathematics, specifically set theory, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. It states:
Or equivalently:
In Zermelo–Fraenkel set theory with the axiom of choice (ZFC), this ...
, the cardinality of
is
, the same as that of
—the set of
real numbers
In mathematics, a real number is a number that can be used to measurement, measure a continuous variable, continuous one-dimensional quantity such as a time, duration or temperature. Here, ''continuous'' means that pairs of values can have arbi ...
.
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, abbreviated AC or AoC, is an axiom of set theory. Informally put, the axiom of choice says that given any collection of non-empty sets, it is possible to construct a new set by choosing one element from e ...
. 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 Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
by using the
order topology
In mathematics, an order topology is a specific topology that can be defined on any totally ordered set. It is a natural generalization of the topology of the real numbers to arbitrary totally ordered sets.
If ''X'' is a totally ordered set, ...
. 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
holds, every increasing ω-sequence of elements of
[0,\omega_1">axiom of countable choice holds, every increasing ω-sequence of elements of /math> converges to a Limit of a sequence">limit in [0,\omega_1">sequence">increasing ω-sequence of elements of [0,\omega_1/math> converges to a Limit of a sequence">limit in [0,\omega_1/math>. 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, is first-countable, 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. More explicitly, a topological space T is second-countable if there exists some countable collection \mat ...
.
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
Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...
.
See also
* Epsilon numbers (mathematics)
In mathematics, the epsilon numbers are a collection of transfinite numbers whose defining property is that they are fixed points of an exponential map. Consequently, they are not reachable from 0 via a finite series of applications of the chos ...
* Large countable ordinal
* Ordinal arithmetic
In the mathematical field of set theory, ordinal arithmetic describes the three usual operations on ordinal numbers: addition, multiplication, and exponentiation. Each can be defined in essentially two different ways: either by constructing an expl ...
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) ...
''. Springer-Verlag, New York, 1978. Reprinted by Dover Publications, New York, 1995. (Dover edition).
{{refend
Ordinal numbers
Topological spaces