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 $s := \langle s_{\alpha}| \alpha < \gamma\rangle$ be a γ-sequence of ordinals. Then s is continuous if at every limit ordinal β < γ,

$s_{\beta} = \limsup\{s_{\alpha}| \alpha < \beta\} = \inf \{ \sup\{s_{\alpha}| \delta \leq \alpha < \beta\} | \delta < \beta\} \,$

and

$s_{\beta} = \liminf\{s_{\alpha}| \alpha < \beta\} = \sup \{ \inf\{s_{\alpha}| \delta \leq \alpha < \beta\} | \delta < \beta\} \,.$

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