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Let $I$ a compact set and $f\in\mathcal{C}([0,1]^I)$. Then exists $J\subset I$, countable or finite, such that:

if $x,y\in [0,1]^I$ such that $x\big|_J=y\big|_J\Rightarrow f(x)=f(y)$.

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    Technical note: the domain is countable-dimensional, not countable.2012-02-22
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    Clearly, if $J$ is a dense subset of $I$, then $x|_J=y|_J$ implies $x=y$, so $f(x)=f(y)$. Therefore, if not true, there cannot be a countable dense subset of $I$. Not sure if that helps. (That assumes that $[0,1]^I$ means the set of continuous functions from $I$ to $[0,1]$, not the set of all functions.)2012-02-22

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