Intuition of product spaces So I have a product space of the form:
$X=X_1 \times \ldots \times X_n$ 
and I take two elements of it, say $x=\{x_1,\ldots,x_n\}$ and $x'=\{x_1',\ldots,x_n'\}$. Now suppose I take the following element:
$x''=x_1,\ldots,x_i',\ldots,x_n$
Can I guarantee that $x'' \in X$? 
 A: Yes. An element of $X$ is just a sequence, the $i$th term of which is from $X_i$ for each $i$. 


*

*Since $x\in X$, we know that the $k$th coordinate of $x''$ (which is $x_k$) is in $X_k$ - for every $k$ except $k=i$.

*However, since $x'\in X$, we know that $x_i'$ is in $X_i$.

*So each of the coordinates of $x''$ are in the corresponding factor space.
It may be more helpful to consider a concrete example: e.g. $X_1=\{4, 5, 6\}$, $X_2=\{7, 8, 9\}$, and $X_3=\{10, 11, 12\}$. Then let $x=(4, 7,10)$, $x'=(5, 8, 11)$, and $i=2$; then $$x''=(4, 8, 10).$$ Do you see why $x''$ is still in $X_1\times X_2\times X_3$?
A: \begin{align*}
x=\{x_1,\ldots,x_n\}\in X\Longrightarrow&\,x_m\in X_m\quad\forall m\in\{1,\ldots,n\}\\
x'=\{x_1',\ldots,x_n'\}\in X\Longrightarrow&\,x_m'\in X_m\quad\forall m\in\{1,\ldots,n\}\\
\hline
\left.\begin{array}{c}x_1\in X_1\\\vdots\\x_i'\in X_i\\\vdots\\x_n\in X_n\end{array}\right\}\Longrightarrow&\,x''=\{x_1,\ldots,x_i',\ldots,x_n\}\in X_1\times\cdots\times X_i\times\cdots\times X_n=X
\end{align*}
