Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 19 / 2 / i / b / Solution

The Bukovský-Hechler theorem states: if $\kappa$ is singular and there are a cardinal $\rho <\kappa$ and a cardinal $\tau$ such that $2^{\mu }=\tau$ for every cardinal $\mu$ with $\rho \le \mu <\kappa$, thenThus an eventual plateau of the power-set function below a singular cardinal continues at that cardinal. One can see the mechanism by decomposing $\kappa$ into $\mathrm{cf}(\kappa )$ bounded pieces: $2^{\kappa }=(2^{<\kappa }{)}^{\mathrm{cf}(\kappa )}$. On the plateau choose $\mu \ge \mathrm{cf}(\kappa )$, so ${\tau }^{\mathrm{cf}(\kappa )}=(2^{\mu }{)}^{\mathrm{cf}(\kappa )}=2^{\mu }=\tau$.