Topology in fractals

In this paper we consider the topological aspects of sets defined by fractal geometry. Topology, as a geometry, is extended to the definition of fractal structures by characterising equivalences in different levels of detail. A group of distinct topological fractals are defined by taking the equipotence and multiplicity properties of sets. Additionally, we define fractal metamorphosis as the sequence obtained by performing a continuous deformation of a fractal structure. In principle, this sequence would include elements of the same topological class. However, by extending the space where fractals are embedded, it is possible to perform a metamorphosis between two fractals belonging to different topological classes when defined in a plane.