Could anyone explain the last statement in this paragraph please:

Let $\displaystyle\prod_{i \in I}A_{i}$ be a Cartesian product. For each $k \in I$ define a map $\displaystyle\pi_k : \prod_{i \in I}A_i \to A_k$ by $f \mapsto f(k)$, or in the other notation $\{a_i\} \mapsto a_k$. $\pi_{k}$ is called (canonical) projection of the product onto its $k$th component (or factor). If every $A_i$ is nonempty, then each $\pi_k$ is surjective (see Excercise 7.6).

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    $\begingroup$ I think your question/confusion might be: if one $A_i$ is empty, then the entire product is empty - in which case the projection onto a factor $A_k$ has a hard time being onto. $\endgroup$ – peter a g Nov 24 '17 at 14:48
  • $\begingroup$ I think that there is also an issue with the scan---the little mark in the last sentence is not a comma; I think that the sentence should read "If every $A_i$ is nonempty...", not "If every $A$, is nonempty..." $\endgroup$ – Xander Henderson Nov 24 '17 at 15:33
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    $\begingroup$ See also this question. $\endgroup$ – Dietrich Burde Nov 24 '17 at 16:07
  • $\begingroup$ yes this is exactly my question @peterag $\endgroup$ – user426277 Nov 24 '17 at 21:11
  • $\begingroup$ @XanderHenderson yes the little mark is not a comma it is an i $\endgroup$ – user426277 Nov 24 '17 at 21:13

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