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A topological map is used to convey information which is not necessarily focused on precise geography. The map is simplified to provide a general overview of an area so that the information on the map is clearer and easier to read. Common examples of a topological map include a subway map, a map included with driving directions to a business, or a map used to represent statistics, such as Internet use in Europe or infant mortality by country. In all of these examples, precise geography takes second place to the information which the map is attempting to communicate.
The London Underground map is perhaps one of the most famous examples of this type of map. The designer, Harry Beck, realized that Tube riders did not need to know precisely where they were, but they did need to be able to see the rough outline of the subway lines. His resulting topological map distorted perspective so that all the lines and stops could clearly be seen. The stylized map is much easier to read. This technique is used on subway maps and route maps around the world, making life much easier for the people trying to use these maps.
Geographical accuracy and scaling are not as important with a topological map. What is important is clearly laying out vital information. Often, this involves stripping a geographical map down to the most vital and basic detail, so that topological information can be laid out over it. This technique is often used to produce maps which are used to convey statistical information about the world, so that people can see roughly where in the world the data is coming from. For example, a table of numbers has a much less profound impact than a map visually demonstrating that the majority of wealth is concentrated in the Northern Hemisphere.
In some cases, these maps may be extremely distorted to emphasize a point or put statistics in context. For example, after an election, a geographical map might tell people that the state of California voted primarily Democratic. A topological map would show that, in fact, Democratic voters are concentrated in certain high population regions of the state, and that in terms of geographical area, most of the state is actually Republican.
A simpler example of this type of map is a map included with the brochure for a business. Most businesses do not provide precise maps laying out every single street in the area. Instead, the map includes major streets and cross streets around the business, so that customers can quickly and easily find it. This simplified map is a topological map designed to provide one piece of information: the best way to get to the business in question.
@Vincenzo -- if you want to get that abstract about it, consider this -- one might say that the "turn by turn" voice navigation featured in your favorite GPS unit or cell phone app is an extension of the logic that gave birth to the topological map. You want those directions to tell you where to point your car and you aren't too interested in landmarks and such that are along the way.
What is really interesting is that maps have been around for years, but people don't get lost as much as they used to when they have accurate, turn-by-turn directions at their fingertips. The need for the topological map is still very much evident today.
Examine a directional map you see on your GPS system, Internet based mapping app or the maps app on your smartphone. You will notice in a hurry that a lot of the features of a topological map have been incorporated into the map on display.
While the map is in scale, the big, red line (or whatever color it is) leading you to your destination is very much in the forefront and is easy to see and follow. The same notion applies to those directional maps as it does to topological maps. People want to use the map to get from point "A" to point "B" and don't give a hang about anything else.
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