It is known that for every continuous map $M:S^1\to \mathbb{R}$ there are infinitely many double points. Also by the Borsuk-Ulam theorem, this is true for each continuous map $N:S^n\to \mathbb{R}^n$, $n\in \mathbb{N}$. A question arising here concerns finding a (connected, metrizable) $k$-manifold $X$ for which each continuous map $S:X \to \mathbb{R}^k$ has a $n$-point, $2<n\in \mathbb{N}$.

A: Does there exist such $X$? and if yes then how it would be figured out?

B: Does there exist a minimal $X$? (means the number of $n$-points of a map $S:X \to \mathbb{R}^k$ be minimum among all $S$ and all $X$.)

Note: a function $f:X\to Y$ has a $n$-point if for some point $p\in Y$ the preimage $f^{-1}(\{p\})$ consists of at least $n$ -points in $X$; a $2$-point is also called a double point.


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