How to change two elements in an uncountably infinite product

Question

I have an uncountable product, say

$$\prod_{i \in I}A_i$$

And i want to replace $A_{i_0}$ and $A_{i_1}$ by $B_{i_0}$ and $B_{i_1}$ respectively. However I know that

$$\left( \prod_{i \in I, \ i \neq i_0, i_1}A_i \right) \times B_{i_0} \times B_{i_1}$$

is not the right way to do this because there is a slight problem with the order of this product.

I could define $C_i$ which equals to $A_i$ for every $i$ except $i_0$ and $i_1$, but I am wondering if there's an easy way to do it.

Answer 1

No, that is the only way to do it. (Your remark on order is true.)

Answer 2

EDIT:

Since the OP qualified he is talking about the product of sets, no such construction as below is possible. This leaves no other option but to define a new sequence.

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Since $I$ is uncountable we would need additional properties for $A_i$ to hold. For example if $I$ is ordered and the products $\prod_{i<i_0} A_i, \prod_{i_0<i<i_1} A_i$ and $\prod_{i>i_1} A_i$ all exist and are finite, you may write what you initially stated: $$P = \prod_{i<i_0} A_i \cdot B_{i_0} \cdot \prod_{i_0<i<i_1} A_i \cdot B_{i_1} \cdot \prod_{i>i_1} A_i = \prod_{i\in I, i\ne i_0, i\ne i_1} A_i \cdot B_{i_0} \cdot B_{i_1} < \infty$$

If we assume the index set was countable and $i_0 < i_1$, You could write $$\prod_{i=1}^{i_0-1} A_i \cdot B_{i_0} \cdot \prod_{i=i_0+1}^{i_1-1} A_i \cdot B_{i_1} \cdot \prod_{i=i_1+1}^\infty A_i$$ Because the first terms are kept in order and constitute a finite product. It is essential that $I$ is countable for this to work. The statement is then also equivalent to $$\prod_{i<i_1, i\ne i_0} A_i \cdot B_{i_0} \cdot B_{i_1} \cdot \prod_{i>i_1} A_i$$ Since the first terms again are finite products.