Write this Cartesian product in math notation

Question

Basically I just want to replace the $\beta$th set in $\displaystyle\prod X_\alpha$ by $U_1$. How would I write the resulting Cartesian product in math notation?

Answer 1

If your index set $I$ is totally ordered, take the product with indices before $\beta$, then $U_1$, then the product with indices after $\beta$.

$$\left(\prod_{\alpha < \beta} X_\alpha \right) \times U_1 \times \left(\prod_{\alpha > \beta} X_\alpha \right)$$

If your index set isn't totally ordered, define $f: \{X_\alpha : \alpha\in I\} \to \{X_\alpha : \alpha\in I\} \cup \{U_1\}$ by

$$ f(X_\alpha) = \begin{cases} X_\alpha & \text{if $\alpha \neq \beta$,} \\ U_1 & \text{if $\alpha = \beta$} \end{cases}. $$

Then the product you want is $\prod_{\alpha \in I} f(X_\alpha)$.

Answer 2

$$U_1×\prod_{\alpha\ne\beta} X_\alpha$$

Answer 3

Your set is exactly $\pi_\beta^{-1}[U_1]$, the set of points whose $\beta$-th coordinate is in $U_1$.