## Transfinite inductions producing coanalytic sets

### Volume 224 / 2014

Fundamenta Mathematicae 224 (2014), 155-174
MSC: Primary 03E15; Secondary 28A05, 03E45, 54H05.
DOI: 10.4064/fm224-2-2

#### Abstract

A. Miller proved the consistent existence of a coanalytic two-point set, Hamel basis and MAD family. In these cases the classical transfinite induction can be modified to produce a coanalytic set. We generalize his result formulating a condition which can be easily applied in such situations. We reprove the classical results and as a new application we show that consistently there exists an uncountable coanalytic subset of the plane that intersects every $C^1$ curve in a countable set.