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One can now go on to define exponentiation of ordinal numbers and explore its properties. Ordinal numbers present an extremely rich arithmetic. There are ordinal numbers which can not be reached from ω with a finite number of the arithmetical operations addition, multiplication and exponentiation. The smallest such is denoted by ε0. ε0 is still countable, but of course there are also uncountable ordinals, the smallest of which is sometimes denoted by Ω. |
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One can now go on to define exponentiation of ordinal numbers and explore its properties. Ordinal numbers present an extremely rich arithmetic. There are ordinal numbers which can not be reached from ω with a finite number of the arithmetical operations addition, multiplication and exponentiation. The smallest such is denoted by ε0. ε0 is still countable, but of course there are also uncountable ordinals, the smallest of which is usually denoted by ω1. |
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In this topology, the sequence 0, 1, 2, 3, 4, ... has limit ω and the sequence ω, ω^ω, ω^(ω^ω), ... has limit ε0. Ordinals which don't have an immediate predecessor can always be written as a limit like this and are called limit ordinals. Ω and its successor Ω+1 are frequently used in topology as the text-book examples of non-countable topologies. For example, in the topological space Ω+1, the element Ω is in the closure of the subset Ω even though no sequence of elements in Ω has the element Ω as its limit. |
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In this topology, the sequence 0, 1, 2, 3, 4, ... has limit ω and the sequence ω, ω^ω, ω^(ω^ω), ... has limit ε0. Ordinals which don't have an immediate predecessor can always be written as a limit like this and are called limit ordinals. The topological spaces ω1 and its successor ω1+1 are frequently used as the text-book examples of non-countable topologies. For example, in the topological space ω1+1, the element ω1 is in the closure of the subset ω1 even though no sequence of elements in ω1 has the element ω1 as its limit. |
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