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 $x'' \in X$? 

\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*}
