Let $A$ be a finite $n$-element set. Let $\mathbb R^A$ be an $n$-dimensional Euclidean space (with the ordinary Euclidean distance). Let $X$ be an arbitrary topological space. Consider a continuous map $f : X\rightarrow \mathbb R^A$.
DEFINITION A point $y\in \mathbb R^A$ is called essential (with respect to $f$) $\Leftarrow:\Rightarrow$ there exists a real $\epsilon > 0$ such that for every continuous $g : X\rightarrow \mathbb R^A$ such that the uniform distance is small: $|g-f| < \epsilon$, point $y$ is a value of $g$, i.e. there exists $x := x_g\in X$ such that $g(x)=y$.
Now consider continuous maps $f:X\rightarrow \mathbb R^B$ $\phi : \mathbb R^B \rightarrow \mathbb R^A$, where $B$ is an $m$-element set, $|B| = m > n = |A|$, and such that the composition $\phi\circ f:X\rightarrow \mathbb R^A$ has an essential value.
QUESTIONS:
- Does there exist a linear map $\lambda : \mathbb R^B\rightarrow \mathbb R^A$ such that $\lambda\circ f: X\rightarrow \mathbb R^A$ has an essential value?
- Does there exist an $n$-element set $C\subset B$ such that $\pi^B_C \circ f:X\rightarrow\mathbb R^C$ has an essential value? (where $\pi^B_C:\mathbb R^B\rightarrow \mathbb R^C$ is the canonical projection).
Here is a special case ($\dim$ stands for the topological dimension): assume that $X\subseteq\mathbb R^B$, and that $\dim(X) \ge n$.
QUESTIONS:
- Does there exist a linear map $\lambda : \mathbb R^B\rightarrow \mathbb R^A$ such that $\lambda|X: X\rightarrow \mathbb R^A$ has an essential value? (we continue to assume that $|A|=n < m$).
- Does there exist an $n$-element set $C\subset B$ such that $\pi^B_C | X: X \rightarrow\mathbb R^C$ has an essential value?
