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So this may be a weird/dumb question, but I was wondering whether a sufficient condition is known for a (countable) family of continuous functions to separate points on, say, a compact metric space $X$ ?

Specifically, given such a $X$ and a continuous function $T : X \rightarrow X$, I was wondering when you can find a continuous function $\phi : X \rightarrow \mathbb{R}$ such that the family $(\phi \circ T^n)_{n \geq 0}$ separates points on $X$ (the intuition is, the more $T$ is mixing, the easier it is ; it cannot work if $T=Id$ and $\dim X \geq 2$, for example).

As far as I know, the property of separating points is more often used to prove something stronger, as in the proof of the Weierstrass theorem, so googling isn't helping.

PS : it is acceptable, if needed, to use finitely many $\phi_i$ instead of just one $\phi$

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