Fixed point

In general mathematics, a fixed point of a function \(f:X\to X\) is any \(x\in X\) such that \(f(x)=x\). If \(f\) is a normal ordinal function, then the fixed points of \(f\) are precisely the closure points of \(f\): that is, for all \(\alpha\), we have \(f(\alpha) = \alpha\) iff, for all \(\beta < \alpha\), \(f(\beta) < \alpha\). Fixed points are useful in the definition and analysis of apeirological notations such as the Veblen hierarchy or OCFs.