  The traditional concept of space in geography is based on the notion of
distance. Where there is a spatial analysis, there is a distance measurement.
However, the precondition for effective distance-based space is that the
geographical systems have characteristic scales. For a scale-free geographical
system, the spatial structure cannot be validly described with pure distance,
and thus the distance-based space is ineffective for geographical modelling. In
the real geographical world, scale-free patterns and processes are everywhere.
We need new notion of geographical space. Using the ideas from fractals and
scaling relations, I propose a dimension-based concept of space for scale-free
geographical analysis. If a geographical phenomenon bears characteristic
scales, we can model it using distance measurement; if a geographical
phenomenon has no characteristic scale, we will describe it using fractal
dimension, which is based on the scaling relations between distance variable
and the corresponding measurements. In short, geographical space fall into two
types: scaleful space and scale-free space. This study shows a new way of
spatial modeling and quantitative analyses for the geographical systems without
characteristic scale.
