In the Fall of 1975 we started a joint project with the ultimate goal of
topo- logically classifying real algebraic sets. This has been a long
happy collaboration (c.f., [K2)). In 1985 while visiting M.S.R.1. we
organized and presented our classification results up to that point in
the M.S.R.1. preprint series [AK14] -[AK17]. Since these results are
interdependent and require some prerequisites as well as familiarity
with real algebraic geometry, we decided to make them self contained by
presenting them as a part of a book in real algebraic geometry. Even
though we have not arrived to our final goal yet we feel that it is time
to introduce them in a self contained coherent version and demonstrate
their use by giving some applications. Chapter I gives the overview of
the classification program. Chapter II has all the necessary background
for the rest of the book, which therefore can be used as a course in
real algebraic geometry. It starts with the elementary properties of
real algebraic sets and ends with the recent solution of the Nash
Conjecture. Chapter III and Chapter IV develop the theory of resolution
towers. Resolution towers are basic topologically defined objects
generalizing the notion of manifold.