Can you elaborate below definition ideally with an example or two? I am starting to read the online book "Algebra and Topology" by Schapira Sierre, but got stuck at the very beginning. He gives the product of a family of sets indexed by $I$, $\left \{ X_i \right \}_{i \in I}$, on page 8 as follows: $$\prod_i X_i := \left \{ \left \{ x_i \right \}_{i \in I} \mid \forall i \in I: x_i \in X_i \right \}$$
Elaborate definition of $\prod_{i} X_i$ where $X_i \in \left \{X_i \right \}_{i \in I}$
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set-theory
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1You can think the product as the set of all functions $f:I\to \cup_i X_i$ such that $f(i)\in X_i$. Can you see what happens if $I=\mathbb{N}$ and $X_i=\mathbb{R}$, for example? – 2017-01-17
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0@Sigur So, for this product, I take one element from each $X_i$ and name it $x_i$, collect them to form $\left \{ x_i \right \}_{i \in I}$. I do this process until I exhaust all possible ways of taking an element from each $X_i$, right? For example, if $X_1$ is a set of all mammal species, $X_2$ is a set of all fruits, and $I = \left \{1,2 \right \}$. Then one element in the product would be $\left \{ monkey_1, apple_2 \right \}$, collect? For your example question, I think the answer is the set of all infinite sequence of real numbers. – 2017-01-18
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0If $I$ is finite, then it is the classical Cartesian product. – 2017-01-18