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** &nbsp; A point $y\in \mathbb R^A$ is called ***essential*** (with respect to $f$) &nbsp; $\Leftarrow:\Rightarrow$ &nbsp; there exists a real $\epsilon > 0$ such that for every continuous &nbsp; $g : X\rightarrow \mathbb R^A$ &nbsp; such that the uniform distance is small: &nbsp; $|g-f| < \epsilon$, &nbsp; point $y$ is a value of $g$, &nbsp; i.e. there exists $x := x_g\in X$ such that $g(x)=y$.

Now consider continuous maps &nbsp; &nbsp; $f:X\rightarrow \mathbb R^B$ &nbsp; &nbsp; $\phi : \mathbb R^B \rightarrow \mathbb R^A$, &nbsp; where $B$  is an $m$-element set, $|B| = m > n = |A|$, &nbsp; and such that the composition &nbsp; $\phi\circ f:X\rightarrow \mathbb R^A$ &nbsp; has an essential value.

**QUESTIONS**:

 1. Does there exist a linear map &nbsp; $\lambda : \mathbb R^B\rightarrow \mathbb R^A$ &nbsp; such that &nbsp; $\lambda\circ f: X\rightarrow \mathbb R^A$ &nbsp; has an essential value?
 2. Does there exist an $n$-element set &nbsp; $C\subset B$ &nbsp; such that &nbsp; $\pi^B_C \circ f:X\rightarrow\mathbb R^C$ &nbsp; has an essential value? &nbsp; (<small>where &nbsp; $\pi^B_C:\mathbb R^B\rightarrow \mathbb R^C$ &nbsp; is the canonical projection</small>).

Here is a special case &nbsp; (<small>$\dim$ &nbsp;stands for the topological dimension</small>): assume that &nbsp; $X\subseteq\mathbb R^B$, &nbsp; and that &nbsp; $\dim(X) \ge n$. 

**QUESTIONS**:

 - Does there exist a linear map &nbsp; $\lambda : \mathbb R^B\rightarrow \mathbb R^A$ &nbsp; such that &nbsp; $\lambda|X: X\rightarrow \mathbb R^A$ &nbsp; has an essential value? &nbsp; &nbsp; (<small>we continue to assume that &nbsp; $|A|=n < m$</small>).
 - Does there exist an $n$-element set &nbsp; $C\subset B$ &nbsp; such that &nbsp; $\pi^B_C | X: X \rightarrow\mathbb R^C$ &nbsp; has an essential value?