# normal (ordinal) function

###### Definition.

A function $F\colon\operatorname{\mathbf{On}}\to\operatorname{\mathbf{On}}$ is continuous if and only if for each $u\subset\operatorname{\mathbf{On}}$ such that $u\neq\emptyset$ it holds that $F(\sup(u))=\sup\{F(\alpha)|\alpha\in u\}$.

###### Definition.

A function $F\colon\operatorname{\mathbf{On}}\to\operatorname{\mathbf{On}}$ is order preserving if and only if for each $\alpha,\beta\in\operatorname{\mathbf{On}}$ such that $\alpha<\beta$ it follows that $F(\alpha).

###### Definition.

A function $F\colon\operatorname{\mathbf{On}}\to\operatorname{\mathbf{On}}$ is a normal function if and only if $F$ is continuous and order preserving.

Title normal (ordinal) function NormalordinalFunction 2013-03-22 15:33:10 2013-03-22 15:33:10 florisje (7763) florisje (7763) 7 florisje (7763) Definition msc 03E10 continuous (for ordinal functions) order preserving (for ordinal functions) normality normal function