Английская Википедия:First uncountable ordinal
Шаблон:Short description In mathematics, the first uncountable ordinal, traditionally denoted by <math>\omega_1</math> or sometimes by <math>\Omega</math>, is the smallest ordinal number that, considered as a set, is uncountable. It is the supremum (least upper bound) of all countable ordinals. When considered as a set, the elements of <math>\omega_1</math> are the countable ordinals (including finite ordinals),[1] of which there are uncountably many.
Like any ordinal number (in von Neumann's approach), <math>\omega_1</math> is a well-ordered set, with set membership serving as the order relation. <math>\omega_1</math> is a limit ordinal, i.e. there is no ordinal <math>\alpha</math> such that <math>\omega_1 = \alpha+1</math>.
The cardinality of the set <math>\omega_1</math> is the first uncountable cardinal number, <math>\aleph_1</math> (aleph-one). The ordinal <math>\omega_1</math> is thus the initial ordinal of <math>\aleph_1</math>. Under the continuum hypothesis, the cardinality of <math>\omega_1</math> is <math>\beth_1</math>, the same as that of <math>\mathbb{R}</math>—the set of real numbers.[2]
In most constructions, <math>\omega_1</math> and <math>\aleph_1</math> are considered equal as sets. To generalize: if <math>\alpha</math> is an arbitrary ordinal, we define <math>\omega_\alpha</math> as the initial ordinal of the cardinal <math>\aleph_\alpha</math>.
The existence of <math>\omega_1</math> can be proven without the axiom of choice. For more, see Hartogs number.
Topological properties
Any ordinal number can be turned into a topological space by using the order topology. When viewed as a topological space, <math>\omega_1</math> is often written as <math>[0,\omega_1)</math>, to emphasize that it is the space consisting of all ordinals smaller than <math>\omega_1</math>.
If the axiom of countable choice holds, every increasing ω-sequence of elements of <math>[0,\omega_1)</math> converges to a limit in <math>[0,\omega_1)</math>. The reason is that the union (i.e., supremum) of every countable set of countable ordinals is another countable ordinal.
The topological space <math>[0,\omega_1)</math> is sequentially compact, but not compact. As a consequence, it is not metrizable. It is, however, countably compact and thus not Lindelöf (a countably compact space is compact if and only if it is Lindelöf). In terms of axioms of countability, <math>[0,\omega_1)</math> is first-countable, but neither separable nor second-countable.
The space <math>[0,\omega_1]=\omega_1 + 1</math> is compact and not first-countable. <math>\omega_1</math> is used to define the long line and the Tychonoff plank—two important counterexamples in topology.
See also
References
Bibliography
- Thomas Jech, Set Theory, 3rd millennium ed., 2003, Springer Monographs in Mathematics, Springer, Шаблон:ISBN.
- Lynn Arthur Steen and J. Arthur Seebach, Jr., Counterexamples in Topology. Springer-Verlag, New York, 1978. Reprinted by Dover Publications, New York, 1995. Шаблон:ISBN (Dover edition).
