Continuous function (set theory)

In 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, if s is an increasing function then 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.

A normal function is a function that is both continuous and strictly increasing.

References

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