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Let $X$ be some set and $\Omega = \prod\limits_{i=0}^\infty X$, so that any element in $\Omega$ can be written as $$ \omega= (\omega_0,\omega_1,\dots) $$ where $\omega_i\in X$, $i \geq 0$. Given $A\subset X$ I wonder what is the right notation for the set $$ A_0 = \{\omega\in \Omega: \omega_0\in A\} $$ I think, the most formal is to write $A_0 = A\times \prod\limits_{i=1}^\infty X$, but I wonder if I can write $A_0 = A\times \Omega$?

I have doubts because if the second notation is formal, then it should mean that $\Omega = X\times \Omega$ - and I am not sure if the latter is correct.

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2 Answers 2

up vote 2 down vote accepted

Neither notation is strictly formal from strictly set-theoretical point of view. An infinite cartesian product is a set of functions from the index set into the union of the producted sets, while a finite cartesian product is usually seen as a set of tuples (which can be seen as ordered pairs of ordered pairs... etc, or just ordered pairs in case of a product of two sets), so if you go down to set-theoretical formalism, they're distinct objects (that is, $A\times \prod_1^\infty X$ is a set of pairs of elements of $X$ and sequences of elements of $X$).

The most formal ay to write it would be just the thing you've written using set-builder notation: $\lbrace \omega\in\Omega\vert \omega_0\in A\rbrace$.

That said, unless you're explicitly dealing with very low-level objects, looking too closely at set-theoretical formalism is neither useful nor enlightening; I think the other two are good enough, if you make it clear what you mean. You might just spell it out in words, or just write something to highlight the abuse like $A\times \Omega\subseteq \Omega$.

Also, an useful notation for cartesian (countable) power is $X^{\mathbf N}$ or $X^{\aleph_0}$ (or even $X^\omega$, but that would be a little too confusing in that context).

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Well, as sets $\Omega\not=X\times\Omega$. However, there are a lot of structures for which there is an isomorphism between $\Omega$ and $X\times\Omega$. For example, in the categories Set, Top, Vect, Grp, et cetera, these objects are isomorphic (at least when the latter is endowed with the usual product structure). If my category theory was better I could probably say more about exactly the categories in which this isomorphism is true. I suspect it might be all categories in which the product always exists, but I'm not sure.

In any case, if you're looking for equality, the correct notation is actually $A_0=\{\omega\in \Omega\mid \omega_0\in A\}$. However, because there is a very natural embedding of $A\times\prod_{i=1}^\infty X_i$ into $\Omega$, the latter notation is widely used and accepted. The notation $A_0=A\times\Omega$ is nonstandard and most people would not correctly interpret the meaning you intended. Remember, as mathematicians we abuse notation all the time, and that's fine, as long as we all abuse it in the same way because the important part is that we understand one another.

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