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 $F(\alpha)<F(\beta)$ .

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
Canonical name	NormalordinalFunction
Date of creation	2013-03-22 15:33:10
Last modified on	2013-03-22 15:33:10
Owner	florisje (7763)
Last modified by	florisje (7763)
Numerical id	7
Author	florisje (7763)
Entry type	Definition
Classification	msc 03E10
Defines	continuous (for ordinal functions)
Defines	order preserving (for ordinal functions)
Defines	normality
Defines	normal function