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What is the formal definition of simple parametric surface?

We are dealing in $\mathbb R^3$. I came across the term in Apostol's Calculus (2nd volume), where he states that the parametric function $\vec r (t)$ is to be one one in the set $T$.

Is the set $T$ inclusive of the boundary? If not, why is the cylinder not a simple parametric surface?

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