# Confusion about general cartesian products [duplicate]

My textbook defined general cartesian products as: Let $$\{A_\alpha| \alpha\in\mathbb{A}\}$$ be a family of sets. The the cartesian product $$\prod_{\alpha} A_\alpha$$ is the set of all maps $$c$$: $$\mathbb{A}\rightarrow\cup_{\alpha} A_\alpha$$ having the property $$\forall\alpha\in\mathbb{A}: c(\alpha)\in A_\alpha$$.

Since the cartesian product $$A*B\neq B*A$$, I was wondering is index set $$\mathbb{A}$$ always has some order to make sure the general product, for example, is $$A_{\alpha 1}*A_{\alpha 2}*A_{\alpha 3}*\dots$$ not $$A_{\alpha 2}*A_{\alpha 1}*A_{\alpha 3}*\dots$$, or the general cartesian product includes both the cases $$A_{\alpha 1}*A_{\alpha 2}*A_{\alpha 3}\dots$$ and $$A_{\alpha 2}*A_{\alpha 1}*A_{\alpha 3}*\dots$$.

It somehow seems in either case the map $$c(\alpha)\in A_\alpha$$ is defined. Thanks!

• You should write $\times$ instead of $*$. – Paul Frost Jan 20 '20 at 23:33
• What is your definition of $A \times B$? – Paul Frost Jan 21 '20 at 0:09

Your book gives the most general definition possible. An useful comparison: $$\Bbb R^n = \prod_{i \in \{1,\ldots,n\}} \Bbb R$$ is the set of functions $$v\colon \{1,\ldots, n\} \to \Bbb R$$ with $$v(i) \in \Bbb R$$ for all $$i$$. We write $$v(i) = v_i$$ and $$v = (v_1,\ldots, v_n)$$. This suggests that you should think of an element $$c \in \prod_{\alpha \in \mathbb{A}} A_\alpha$$ as a vector with $$|\mathbb{A}|$$ components (even if $$|\mathbb{A}|$$ is infinite!), one for each factor $$A_\alpha$$.
As far as order go, you are correct. If $$A_1 = A$$ and $$A_2 = B$$, both $$A\times B$$ and $$B\times A$$ are equal to $$\prod_{\alpha \in \{1,2\}} A_\alpha$$, but choosing an order $$A\times B$$ or $$B\times A$$ to write requires a choice of ordering for the index set $$\mathbb{A} = \{1,2\}$$ (either $$1<2$$ or $$2<1$$).