What theorem in topology was Claude Shannon referring to

Question

In 1949, in a classic paper, Claude Shannon$^\color{red}{\star}$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?

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_{$\color{red}{\star}$ 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.}

Answer 1

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 $\mathbb{R}^m$.