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

$s_{\beta }=\mathrm{lim\; sup}\{s_{\alpha }|\alpha <\beta \}=\inf \{\sup \{s_{\alpha }|\delta \le \alpha <\beta \}|\delta <\beta \}$

and

$s_{\beta }=\mathrm{lim\; inf}\{s_{\alpha }|\alpha <\beta \}=\sup \{\inf \{s_{\alpha }|\delta \le \alpha <\beta \}|\delta <\beta \}.$

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