# What theorem in topology was Claude Shannon referring to?

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?

• Some of the ideas are explained in en.wikipedia.org/wiki/Space-filling_curve – reuns Mar 2 at 20:14
• 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. – Lee Mosher Mar 2 at 20:31

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 $$m, we get as a corollary that there is no continuous injection from a (nonempty) open subset of $$\mathbb{R}^n$$ to $$\mathbb{R}^m$$.