First uncountable ordinal
Encyclopedia
In mathematics
, the first uncountable ordinal, traditionally denoted by ω1 or sometimes by Ω, is the smallest ordinal number
that, considered as a set, is uncountable. It is the supremum
of all countable ordinals. The elements of ω1 are the countable ordinals, of which there are uncountably many.
Like any ordinal number (in von Neumann's approach), ω1 is a well-ordered set
, with set membership ("∈") serving as the order relation. ω1 is a limit ordinal, i.e. there is no ordinal α with α + 1 = ω1.
The cardinality of the set ω1 is the first uncountable cardinal number
, ℵ1 (aleph-one). The ordinal ω1 is thus the initial ordinal of ℵ1.
Indeed, in most constructions ω1 and ℵ1 are 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. (See Hartogs number
.)
by using the order topology
. When viewed as a topological space, ω1 is often written as [0,ω1) to emphasize that it is the space consisting of all ordinals smaller than ω1.
Every increasing ω-sequence
of elements of [0,ω1) converges to a limit
in [0,ω1). The reason is that the union
(=supremum) of every countable set of countable ordinals is another countable ordinal.
The topological space [0,ω1) is sequentially compact but not compact
. It is however countably compact
and thus not Lindelöf
. In terms of axioms of countability, [0,ω1) is first countable but not separable nor second countable. As a consequence, it is not metrizable.
The space [0, ω1] = ω1 + 1 is compact and not first countable. ω1 is used to define the long line
and the Tychonoff plank
, two important counterexamples in topology
.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the first uncountable ordinal, traditionally denoted by ω1 or sometimes by Ω, is the smallest ordinal number
Ordinal number
In set theory, an ordinal number, or just ordinal, is the order type of a well-ordered set. They are usually identified with hereditarily transitive sets. Ordinals are an extension of the natural numbers different from integers and from cardinals...
that, considered as a set, is uncountable. It is the supremum
Supremum
In mathematics, given a subset S of a totally or partially ordered set T, the supremum of S, if it exists, is the least element of T that is greater than or equal to every element of S. Consequently, the supremum is also referred to as the least upper bound . If the supremum exists, it is unique...
of all countable ordinals. The elements of ω1 are the countable ordinals, of which there are uncountably many.
Like any ordinal number (in von Neumann's approach), ω1 is a well-ordered set
Well-order
In mathematics, a well-order relation on a set S is a strict total order on S with the property that every non-empty subset of S has a least element in this ordering. Equivalently, a well-ordering is a well-founded strict total order...
, with set membership ("∈") serving as the order relation. ω1 is a limit ordinal, i.e. there is no ordinal α with α + 1 = ω1.
The cardinality of the set ω1 is the first uncountable cardinal number
Cardinal number
In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality of sets. The cardinality of a finite set is a natural number – the number of elements in the set. The transfinite cardinal numbers describe the sizes of infinite...
, ℵ1 (aleph-one). The ordinal ω1 is thus the initial ordinal of ℵ1.
Indeed, in most constructions ω1 and ℵ1 are 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. (See Hartogs number
Hartogs number
In mathematics, specifically in axiomatic set theory, a Hartogs number is a particular kind of cardinal number. It was shown by Friedrich Hartogs in 1915, from ZF alone , that there is a least well-ordered cardinal greater than a given well-ordered cardinal.To define the Hartogs number of a set it...
.)
Topological properties
Any ordinal number can be turned into a topological spaceTopological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
by using the order topology
Order topology
In mathematics, an order topology is a certain 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...
. When viewed as a topological space, ω1 is often written as [0,ω1) to emphasize that it is the space consisting of all ordinals smaller than ω1.
Every increasing ω-sequence
Sequence
In mathematics, a sequence is an ordered list of objects . Like a set, it contains members , and the number of terms is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence...
of elements of [0,ω1) converges to a limit
Limit of a sequence
The limit of a sequence is, intuitively, the unique number or point L such that the terms of the sequence become arbitrarily close to L for "large" values of n...
in [0,ω1). The reason is that the union
Union (set theory)
In set theory, the union of a collection of sets is the set of all distinct elements in the collection. The union of a collection of sets S_1, S_2, S_3, \dots , S_n\,\! gives a set S_1 \cup S_2 \cup S_3 \cup \dots \cup S_n.- Definition :...
(=supremum) of every countable set of countable ordinals is another countable ordinal.
The topological space [0,ω1) is sequentially compact but not compact
Compact space
In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
. It is however countably compact
Countably compact space
In mathematics a topological space is countably compact if every countable open cover has a finite subcover.-Examples and Properties:A compact space is countably compact...
and thus not Lindelöf
Lindelöf space
In mathematics, a Lindelöf space is a topological space in which every open cover has a countable subcover. The Lindelöf property is a weakening of the more commonly used notion of compactness, which requires the existence of a finite subcover....
. In terms of axioms of countability, [0,ω1) is first countable but not separable nor second countable. As a consequence, it is not metrizable.
The space [0, ω1] = ω1 + 1 is compact and not first countable. ω1 is used to define the long line
Long line (topology)
In topology, the long line is a topological space somewhat similar to the real line, but in a certain way "longer". It behaves locally just like the real line, but has different large-scale properties. Therefore it serves as one of the basic counterexamples of topology...
and the Tychonoff plank
Tychonoff plank
In topology, the Tychonoff plank is a topological space that is a counterexample to several plausible-sounding conjectures. It is defined as the product of the two ordinal space[0,\omega_1]\times[0,\omega]...
, two important counterexamples in topology
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
.
Reference
- Thomas Jech, Set Theory, 3rd millennium ed., 2003, Springer Monographs in Mathematics, Springer, ISBN 3-540-44085-2.
- Lynn Arthur Steen and J. Arthur Seebach, Jr., Counterexamples in TopologyCounterexamples in TopologyCounterexamples in Topology 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 have defined a wide variety of topological properties...
. Springer-Verlag, New York, 1978. Reprinted by Dover Publications, New York, 1995. ISBN 0-486-68735-X (Dover edition).