  Fractal geometry proved to be an effective mathematical tool for exploring
real geographical space based on digital maps and remote sensing images.
Whether the fractal theory tool can be applied to abstract geographical space
has not been reported. An abstract space can be defined by multivariable
distance metrics, which is frequently met in scientific research. Based on the
ideas from fractals, this paper is devoted to developing correlation dimension
analysis method for generalized geographical space by means of mathematical
derivation and empirical analysis. Defining a mathematical distance or
statistical distance, we can construct a generalized correlation function. If
the relationship between correlation function and correlation lengths follows a
power law, the power exponent can be demonstrated to associate with correlation
dimension. Thus fractal dimension can be employed to analyze the structure and
nature of generalized geographical space. This suggests that fractal geometry
can be generalized to explore scale-free abstract geographical space. The
theoretical model was proved mathematically, and the analytical method was
illustrated by using observational data. This research is helpful to expand the
application of fractal theory in geographical analysis, and the results and
conclusions can be extended to other scientific fields.
