  One of the important consequences of the Banach Fixed Point Theorem is
Hutchinson's theorem which states the existence and uniqueness of fractals in
complete metric spaces. The aim of this paper is to extend this theorem for
semimetric spaces using the results of Bessenyei and P\'ales published in 2017.
In doing so, some properties of semimetric spaces as well as of the fractal
space are investigated. We extend Hausdorff's theorem to characterize
compactness and Blaschke's theorems to characterize the completeness of the
fractal space. Based on these preliminaries, an analogue of Hutchinson's
Theorem in the setting of semimetric spaces is proved and finally, error
estimates and stability of fractals are established as well.
