Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.
This question is aimed to provide a completely new proof of Brouwer's fixed point theorem. So, a possible positive answer should be independent of any result based on Brouwer's theorem. To prove the required density, it would be reasonable to use the Baire category theorem.
