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Is there a topological space $X$, which is not a singleton, and satisfies the following property?

For every continuous function $f: Xtimes S^2tomathbbR^2$ there exist a point $xin S^2$ such that $f(t, x)=f(t,-x),;forall t in X$?

Is there a classification of all spaces $X$ with such property?

Theorem. For a topological space $X$ the following conditions are equivalent:

1) for any continuous map $f:Xtimes S^2tomathbb R^2$ there exists a point $sin S^2$ such that $f(x,s)=f(x,-s)$ for any $xin X$;

2) any continuous map $f:Xto mathbb R$ is constant.

Proof. (1) $Rightarrow$ (2) Assume that $X$ admits a non-constant map $hbar :Xto mathbb R$. We lose no generality assuming that $0,1subset hbar(X)subset[0,1]$.
Then there are points $x_0,x_1in X$ such that $hbar(x_i)=i$ for $iin0,1$.

Let $p:S^2tomathbb R^2$, $p:(x,y,z)mapsto (x,y)$, be the projection of the sphere $S^2=(x,y,z)inmathbb R^3:x^2+y^2+z^2=1$ onto the plane.

Let $varphi:S^2to S^2$ be any homeomorphism of the sphere $S^2$ such that $pcirc varphi(0,0,1)ne pcirc varphi(0,0,-1)$.

Using the Tietze-Urysohn Theorem, find a continuous map $psi:[0,1]times S^2tomathbb R^2$ such that $psi(0,s)=p(s)$ and $psi(1,x)=pcircvarphi(s)$ for all $sin S$. Then the continuous map $$f:Xtimes S^2tomathbb R^2,;;f:(x,s)mapsto psi(hbar(x),s),$$ has the following property:

if $f(x_0,s)=f(x_0,-s)$ for some $sin S^2$, then $sin(0,0,1),(0,0,-1)$ and $f(x_1,s)ne f(x_1,-s)$.

(2) $Rightarrow$ (1) Assume that each continuous map $Xtomathbb R$ is constant and take any continuous function $f:Xtimes S^2tomathbb R^2$. Fix any point $x_0in X$ and using the Borsuk-Ulam Theorem, find a point $sin S^2$ such that $f(x_0,s)=f(x_0,-s)$.
By our assumption, for every $sin S^2$ the function $frestrictionXtimess$ is constant. So, for every $xin X$ we have
$$f(x,s)=f(x_0,s)=f(x_0,-s)=f(x,-s).$$

Remark. For examples of regular topological spaces on which all continuous real-valued functions are constant, see page 119 of Engelking's "General Topology".