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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:

  1. 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?
  2. 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?
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The notion of an essential value was introduced by Pavel S. Alexandrov (right?), who proved that a space has dimension at least $n\ \ \Leftrightarrow$ there exists a continuous map $f : X \rightarrow \mathbb R^n$ which has an essential value (I think it was him, at least for metric compact spaces or separable metric spaces). –  Wlodzimierz Holsztynski May 3 '13 at 3:49
    
Including phrase "implicit function" would make the title of my question more attractive (attractiver :-) but it would force me into too much of explaining within the body of the question that it is so and so but not quite. –  Wlodzimierz Holsztynski May 3 '13 at 3:52

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