Title: Special sets of reals
Author: Arnold W. Miller
Abstract:
This is a survey paper which will update chapter 5 of the
Handbook of set theoretic topology, edited by Kunen and Vaughan,
North-Holland, 1984. It is concerned with totally imperfect subsets
of the reals. A set of reals is totally imperfect iff
it does not contain an uncountable closed set.
Examples of such sets are:
Luzin sets, Sierpinski sets, concentrated sets, strong measure zero sets,
universal measure zero sets, perfectly meager sets,
strong first category sets, sigma-sets, lambda-sets, Q-sets,
gamma-sets, s_0-sets, and C''-sets.
We are primarily concerned with techniques of construction.
The classical method is transfinite induction using the continuum hypothesis.
Most of these arguments are easily generalized to use Martin's axiom.
Some more exotic set theory has also been used: Diamond,
Aronszajn trees, Ulam matrices, and forcing.
In some forcing arguments we start with a set of reals X in a model of
set theory M. Then we construct a generic extension N of M. In N
the properties of set X may change even though X is the same
set of reals.