Brouwer Fixed Point Theorem and Invariance of Domain Theorem.

I would like to know references containing proofs of equivalence (or an implication in any direction) between Brouwer Fixed Point Theorem and Invariance of Domain Theorem.

Evidence that makes me believe that they are equivalent is this post in the blog of Terence Tao.But the text of this blog does not provide any bibliographical reference. Just a comment.

I looked for this equivalence (or an implication in any direction) in several classic books of Analysis:

But I did not find anything.An encyclopedic book that shows several theorems equivalent to Brower's fixed-point theorem is Nonlinear Functional Analysis and its Applications: IV: Applications to Mathematical Physics by E. Zeidler. On page 795 of this book more than a dozen theorems equivalent to Brower's fixed-point theorem are shown. But no mension to the domain invariance theorem is made.

Thanks in advance for any bibliography that can help me.

Brouwer Fixed Point Theorem. Let $f:U\to\mathbb{R}^n$ be a continuous function defined in an open set $U\subset\mathbb{R}^n$. Let $K\subset U$ convex and compact. If $f(K)\subset K$ then there is $x_0\in K$ such that $f(x_0)=x_0$.

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Invariance of Domain Theorem. Let $f:U\to\mathbb{R}^n$ be a continuous function defined in an open set $U\subset\mathbb{R}^n$. If $f$ is one to one function then $f(U)$ is a open subset of $\mathbb{R}^n$.

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• I believe that many books in algebraic topology contain at least one direction of the result. Maybe Elements of algebraic topology by Munkres is one of them. – asatzhh Jul 19 '18 at 14:27
• To some extent this is a philosophical question: All true results may be considered as equivalent from an abstract logical point of view. But you are interested to derive each theorem as a corollary from the other using an "easy" transformation. The usual proofs which you find in the modern literature in my opinion do not accomplish this, they only use a common method (e.g. homology theory) to prove both results. Perhaps math.stackexchange.com/q/2414911 helps a little. – Paul Frost Jul 19 '18 at 16:22
• math.stackexchange.com/q/2723272 – Paul Frost Jul 28 '18 at 14:15