Non-Normal Magidor-Radin Types of Forcings

We develop the non-normal variations of two classical Prikry-type forcings; namely, Magidor and Radin forcings. We generalize the fact that the non-normal Prikry forcing is a projection of the extender-based to a coordinate of the extender to our forcing and the Radin/Magidor-Radin-extender-based forcing from \cite{CarmiMagidorRadin,CarmiRadin}. Then, we show that both the non-normal variation of Magidor and Radin forcings can add a Cohen generic function to every limit point of cofinality $\omega$ of the generic club. Second, we show that this phenomenon is limited to the cases where the forcings are not designed to change the cofinality of a measurable $\kappa$ to $\omega_1$. Specifically, in the above-mentioned circumstances these forcings do not project onto any $\kappa$-distributive forcing. We use that to conclude that the extender-based Radin/Magidor-Radin forcing does not add fresh subsets to $\kappa$ as well. In the second part of the paper we focus on the natural non-normal variation of Gitik's forcing from \cite[\S3]{GitikNonStationary}. Our main result shows that this poset can be employed to change the cofinality of a measurable cardinal $\kappa$ to $\omega_1$ while introducing a Cohen subset of $\kappa$.