  In this survey, my aim has been to discuss the use of sequences and countable
sets in general topology. In this way I have been led to consider five
different classes of topological spaces: first countable spaces, sequential
spaces, Frechet spaces, spaces of countable tightness and perfect spaces. We
are going to look at how these classes are related, and how well the various
properties behave under certain operations, such as taking subspaces, products,
and images under proper mappings. Where they are not well behaved we take the
opportunity to consider some relevant examples, which are often of special
interest. For instance, we examine an example of a Frechet space with unique
sequential limits that is not Hausdorff. I asked the question of whether there
exists in ZFC an example of a perfectly normal space that does not have
countable tightness: such an example was supplied and appears below. In our
discussion we shall report two independence theorems, one of which forms the
solution to the Moore-Mrowka problem. The results that we prove below include
characterisation theorems of sequential spaces and Frechet spaces in terms of
appropriate classes of continuous mappings, and the theorem that every
perfectly regular countably compact space has countable tightness.
