While surfing the web the other day I read an article in which the author refers to a “topological map.” I think it is safe to say that he meant to write “topographic map.” This is an error I’ve seen many times before.
A topographic map is a map of a region that shows changes in elevation, usually with contour lines indicating different fixed elevations. This is a map that you would take on a hike.
A topological map is a continuous function between two topological spaces—not the same thing as a topographic map at all!
I thought for sure that there was no cartographic meaning for topological map. It turns out, however, that there is.
A topological map is a map that is only concerned with relative locations of features on the map, not on exact locations. A famous example is the graph that we use to solve the Bridges of Königsberg problem.
Another famous class of topological maps are subway maps (and other transportation maps in which the vehicles travel along fixed routes). In a subway map, what is most important are the orders of the subway stops on each line and the interconnections between the various subway lines. The map of the London Underground is a well-known topological map.
I’m very happy to learn of this definition. It is a perfect use of the term topological!
(Having said all of this, I still think the author meant topographic map.)
David Richeson: Division by Zero 
Nice tidbit of information. The topological map of a subway system is a very useful alternative to spatial mapping. Often, removal of one type of information from a data-set can facilitate the utility of the retained data. In this case distance between points and low frequency directional data were sacrificed to improve usability.
I used to transform massive seismic data sets to a simpler form (essentially a 3D topological map of the data) for interpretation/mapping then reverse the transformation and apply to the new topologic maps resulting in a topographic map of a subsurface structure.
Do you know of other practical examples of using topological maps in engineering or research?
There is a lot of research these days for topological methods for computer network routing and robotic control.
Often no humans are involved, so no maps per se, but they could have them. Computer networks are nearly always mapped topologically.
Topological maps provide the basis for graph theory. Converting a topographic map into a topological map enables the use of a large tool kit.
In many way a mind map or conceptual map of non-physical entities can be converted into a spatial map. We already do this when we ask questions like how many years will it be before technology x enables us to do task y?
I’ve actually created product roadmaps that leverage both temporal and spatial relationships. These become topographic maps where distant events are rendered as topological maps.