In 1949, in a classic paper, Claude Shannon wrote the following:
As we change the message a small amount, the corresponding signal will change a small amount, until some critical value is reached. At this point the signal will undergo a considerable change. In topology it is shown that it is not possible to map a region of higher dimension into a region of lower dimension continuously. It is the necessary discontinuity which produces the threshold effects we have been describing for communication systems.
Emphasis is Shannon's. I assume Shannon is referring to a very well-known theorem in topology. Can anyone tell me which theorem that is?
- Claude Elwood Shannon, Communication in the Presence of Noise, Proceedings of the Institute of Radio Engineers, Volume 37, Issue 1, Pages 10-21, January 1949.