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?

  • $\begingroup$ Some of the ideas are explained in en.wikipedia.org/wiki/Space-filling_curve $\endgroup$ – reuns Mar 2 at 20:14
  • $\begingroup$ Just from the wording, it looks like it is saying that a nonempty open subset of $\mathbb R^n$ cannot be mapped continuously and injectively (in)to $\mathbb R^k$ if $k < n$. I think that the proof is not hard, and just uses what nowadays is first year algebraic topology. $\endgroup$ – Lee Mosher Mar 2 at 20:31

This sounds like invariance of domain. This is usually stated something like:

If $U\subseteq\mathbb{R}^n$ is open and $f:U\rightarrow\mathbb{R}^n$ is continuous and injective, then $f$ is a homeomorphism between $U$ and its image.

In particular, the image of any injective continuous map $\mathbb{R}^n\rightarrow\mathbb{R}^n$ is locally homeomorphic to $\mathbb{R}^n$. Via the usual inclusion map $\mathbb{R}^m\rightarrow\mathbb{R}^n$ for $m<n$, we get as a corollary that there is no continuous injection from a (nonempty) open subset of $\mathbb{R}^n$ to $\mathbb{R}^m$.

  • $\begingroup$ Thank you. Would you say that the algebraic topology tag is applicable? $\endgroup$ – Rodrigo de Azevedo Mar 2 at 20:17
  • $\begingroup$ @RodrigodeAzevedo I think it's a borderline case - I'd neither object to nor demand it, personally. $\endgroup$ – Noah Schweber Mar 2 at 20:18
  • $\begingroup$ But the general topology tag is OK, right? I did not even know how to tag the question properly. $\endgroup$ – Rodrigo de Azevedo Mar 2 at 20:19
  • 2
    $\begingroup$ @RodrigodeAzevedo invariance of domain is definitely general topology (first proved by Brouwer, far before algebraic topology had been developed (his proof did inspire some techniques we now associate with it). $\endgroup$ – Henno Brandsma Mar 2 at 20:22

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