  Much of the structure in metric spaces that allows for the creation of
fractals exists in more generalized non-metrizable spaces. In particular the
same theorems regarding the behavior of compact sets can be proven in the more
general framework of $\beta$-spaces. However in most $\beta$-spaces, a set
being compact (and more generally being totally bounded) is so restrictive as
to render all fractal examples completely uninteresting. In this paper we
provide a generalization of compact sets, continuous functions, and all the
related machinery necessary for fractals to be defined as the unique fixed set
of an IFS. We conclude by discussing some interesting examples of
non-metrizable fractals.
