Ordinal numbers | Topological spaces

First uncountable ordinal

In mathematics, the first uncountable ordinal, traditionally denoted by or sometimes by , 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 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, (aleph-one). 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. For more, see Hartogs number. (Wikipedia).

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Related pages

Topological space | Counterexamples in Topology | Hartogs number | Separable space | Topology | Order topology | First-countable space | Long line (topology) | Limit ordinal | Metrizable space | Second-countable space | Lindelöf space | Sequence | Ordinal number | Well-order | Large countable ordinal | Ordinal arithmetic | Countably compact space | Cardinal number | Mathematics | Set (mathematics) | Continuum hypothesis | Union (set theory) | Real number | Limit of a sequence | Compact space | Tychonoff plank | Cardinality