Determine if $\prod\limits_{i=0}^\infty A_i$ countable or uncountable

Question

Let $A_i$ be a family of non-empty sets for every $i \in N$

Determine if $\prod\limits_{i=0}^\infty A_i$ is always countable, some times countable, or never countable.

a) $|A_i| \ge 2$ for each $i \in \mathbb{N}$

$A$ is the set of all infinite sequences whereby $A=\{(a_0,a_1,a_2,\ldots) \mid a_k \in A_k \forall k \in \mathbb {N} \}$. To show that $A$ is uncountable, it suffices to show that $\not \exists$ bijection $f: \mathbb{N} \to A$. Assume for contradiction that $\exists$ bijection $f: \mathbb{N} \to A$.

Note that $f(0)=(a_0^{(0)},a_1^{(0)},a_2^{(0)},\ldots)$

and that $f(1)=(a_0^{(1)},a_1^{(1)},a_2^{(1)},\ldots)$

This means that $f(K)=(a_0^{(k)},a_1^{(k)},a_3^{(k)},\ldots)$. So each natural number is mapped to some infinite sequence. To show that $f$ is not surjective, let $b \in A \ni b=(b_0, b_1, b_2, b_3, \ldots)$ where $b_k \in A_k \forall k \in \mathbb{N}$.

Since $|A_0| \ge 2, |A_1| \ge 2, |A_2| \ge 2,\ldots,|A_i| \ge i$, let $b_i \in A_i \not \{a_i^{(i )}\}$ for every $i \in \mathbb{N}$. So the i-th term of $b$ will be different from the i-th term of $f$, and hence $b \notin f(\mathbb{N})$. Thus we have $b \in A \ni f(i) \ne b \forall i \in \mathbb{N}$. However, this contradicts the premise that $f$ is surjective (i.e. $f(\mathbb{N})=A)$. Therefore, $A$ is uncountable.

b) $\exists N \in \mathbb{N} \ni \forall i > N$, $|A_i| \ge 2$.

I'm having trouble figuring out what this means. Does this mean $|A_1| \ge 2, |A_2| \ge 2, |A_3| \ge 2, \ldots,|A_n| \ge 2$ for every $n \in \mathbb{N}$ except $n=0$? I'm thinking that in this case, the zeroth term of $b$ will be the zeroth term of $f$, so $f(0)=b$. But $f$ may not be injective since $f(1)=b$, but $0 \ne 1$. So it will essentially be some times countable.

c) $\exists N \in \mathbb{N} \ni \forall i > N$, $|A_i| = 1$.

As $f(K)=(a_0^{(k)},a_1^{(k)},a_3^{(k)},\ldots)$, where each $a_i^{(k)} \in A_i$ for every $k \in \mathbb{N}$ and $|A_n|=1$. This means that $f$ cannot be surjective since every natural number maps to $b$. Thus $\prod\limits_{i=0}^n A_n$ is uncountable.

However, $\prod\limits_{i=1}^{n}A_n$ is countable since it is finite; that is, $\left|\prod\limits_{i=1}^n A_n\right|=|A_1| \times |A_2| \times |A_3| \times\cdots\times |A_n|=1$?

d) $\exists N \in \mathbb{N} \ni \forall n > N$, $|A_n| = 1$, and $\exists$ injection $f:\mathbb{N} \to A$.

I'm not sure about this. My understanding is that $A$ should infinite since by definition, a set $A$ is infinite if there exists an injection $f: \mathbb{N} \to B$.

Any insight would be appreciated. Thanks!

Answer 1

Your proof of part (a) is correct, a nice application of diagonal argument.

For part (b), you can consider it as follows: $$\left|\prod_{i=0}^{\infty}{A_i}\right|=\left|\prod_{i=0}^{N}{A_i}\right|\cdot\left|\prod_{i=N+1}^{\infty}{A_i}\right|$$

You argument in part (a) has already shown that the second term on the RHS is uncountable. Since each $A_i$ is nonempty, the whole product must be uncountable. To illustrate this idea clearly, you can take the argument given by @user2628206 in the comment.

For part (c), the term $\displaystyle\prod_{i=N+1}^{\infty}{A_i}$ is a singleton since it is assumed that $|A_i|=1$ for $i>N$, so $$\left|\prod_{i=0}^{\infty}{A_i}\right|=\left|\prod_{i=0}^{N}{A_i}\right|=\prod_{i=0}^{N}{|A_i|}.$$ The answer is then undetermined:

1. If each $A_i$ is at most countable (i.e., finite or countably infinite), then the product is at most countable.

2. If there is one uncountable $A_i$, then the product is uncountable, e.g., $\Bbb R\times\Bbb Q$.

The only difference between part (d) and part (c) is the injection $f:\Bbb N\to A$. This suggests that $A$ must be infinite. Thus again,

1. If no $A_i$ is uncountable, then the product is countably infinite;

2. If at least one $A_i$ is uncountable, then the product is uncountable.