> "Topology ignores such things as corners, distances, and convexity."
No, that's what topology is.
> "That makes it hard to combine with GIS systems."
No, GIS systems rely heavily on topology. Here are two definitions:
Topology: 1. the study of geometric properties and spatial relations unaffected by the continuous change of shape or size of figures.
Topography: 1. the arrangement of the natural and artificial physical features of an area.
- basic topology like detecting whether or not a point lies within a complex polygon
- topological morphology, such as smoothing a line or polygon while maintaining boundaries (this is probably the most common definition in the GIS world)
- topological interpolation, such as estimating z values over a 3d mesh[1][2] (there are a number of methodologies here, and this represents probably the most interesting intersection of topology and statistics)
Also note that turf is geared towards GIS, but most of the underlying operations are not GIS specific and apply to geometry more generally. You will find many of the same algorithms in applications like game engines, for example.
[1] http://en.wikipedia.org/wiki/Kriging [2] http://en.wikipedia.org/wiki/Contour_line
If you are groking topology in the mathematical sense, you'd be talking about concepts like (co)homology and homotopy groups, and be doing quite a lot of commutative algebra. You definitely would not care about statistics and point location.
My library is definitely not a topology library. I tend to describe it as a geospatial processing engine. My first comment was really only meant to speak to some overlap I noticed with the topology described in the article, and to convey my deep respect for the people who have a more full understanding of the low level mathematics involved (which it sounds like you might possess).
While many would classify the GP's library as one primarily concerned with computational geometry (but certainly not topography), there are large overlaps of computational geometry with computational topology. Many topology results won't be useful in an applied field like GIS, but other results will be extremely useful as they work regardless of the mess of real world measurements.
Classification of branches of mathematics is a losing game, anyway. You try to separate two bodies of work, you'll invariably find someone's extensive results that straddle the boundary.