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Continuous function (set theory)
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Continuous function (set theory) : ウィキペディア英語版
Continuous function (set theory)
In mathematics, specifically set theory, a continuous function is a sequence of ordinals such that the values assumed at limit stages are the limits (limit suprema and limit infima) of all values at previous stages. More formally, let γ be an ordinal, and s := \langle s_| \alpha < \gamma\rangle be a γ-sequence of ordinals. Then ''s'' is continuous if at every limit ordinal β < γ,
:s_ = \limsup\ = \inf \ | \delta < \beta\} \,
and
:s_ = \liminf\ = \sup \ | \delta < \beta\} \,.
Alternatively, ''s'' is continuous if ''s'': γ → range(s) is a continuous function when the sets are each equipped with the order topology. These continuous functions are often used in cofinalities and cardinal numbers.
==References==

* Thomas Jech. ''Set Theory'', 3rd millennium ed., 2002, Springer Monographs in Mathematics,Springer, ISBN 3-540-44085-2

抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)
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