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 α | α < γ ⟩ {\displaystyle 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 β = lim sup { s α : α < β } = inf { sup { s α : δ ≤ α < β } : δ < β } {\displaystyle s_{\beta }=\limsup\{s_{\alpha }:\alpha <\beta \}=\inf\{\sup\{s_{\alpha }:\delta \leq \alpha <\beta \}:\delta <\beta \}} and s β = lim inf { s α : α < β } = sup { inf { s α : δ ≤ α < β } : δ < β } . {\displaystyle s_{\beta }=\liminf\{s_{\alpha }:\alpha <\beta \}=\sup\{\inf\{s_{\alpha }:\delta \leq \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.

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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 α | α < γ ⟩ {\displaystyle 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 β = lim sup { s α : α < β } = inf { sup { s α : δ ≤ α < β } : δ < β } {\displaystyle s_{\beta }=\limsup\{s_{\alpha }:\alpha <\beta \}=\inf\{\sup\{s_{\alpha }:\delta \leq \alpha <\beta \}:\delta <\beta \}}

and

s β = lim inf { s α : α < β } = sup { inf { s α : δ ≤ α < β } : δ < β } . {\displaystyle s_{\beta }=\liminf\{s_{\alpha }:\alpha <\beta \}=\sup\{\inf\{s_{\alpha }:\delta \leq \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

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

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