Is proximity preservation rational in social choice theory?

We establish a strong impossibility theorem of a rational social choice that the proximity preservation (also called weak proximorphism WPX) and the diagonal surjectivity are logically inconsistent. The result is valid for finite or infinite alternatives, discrete or continuous. It generalizes the Baigent theorem, largely weakening his antecedent. For continuum set of alternatives, we clarify the notion of WPX by showing (1) WPX almost implies the continuity, (2) WPX is almost rigid. These observations raise the issue whether WPX is a natural condition for a social welfare function. A splitting reformulation of the proximity preservation which is weaker but rational is suggested.