Which of the following subsets of \(\mathbb{R}^{\omega}\) can be expressed as the cartesian product of subsets of \(\mathbb{R}\) ?
\(\{x \mid x_{i} \text{ is an integer for all } i\}\) .
\(\{x \mid x_{i} \geq i \text{ for all } i\}\) .
\(\{\mathbf{x} \mid x_i\) is an integer for all \(i \geq 100\}\) .
\(\{x \mid x_{2} = x_{3}\}\) .
Solution
Yes. The set is \[ \prod_{i=1}^{\infty}\mathbb Z. \]
Yes. It is \[ \prod_{i=1}^{\infty}[i,\infty). \]
Yes. It is \[ \left(\prod_{i=1}^{99}\mathbb R\right) \times \left(\prod_{i=100}^{\infty}\mathbb Z\right). \] Equivalently, take the \(i\)-th factor to be \(\mathbb R\) for \(i<100\) and \(\mathbb Z\) for \(i\ge100\).
No. Suppose \[ \{x:x_2=x_3\}=\prod_i A_i. \] For every \(r\in\mathbb R\) there is a sequence in the set having \(x_2=x_3=r\), so necessarily \[ A_2=A_3=\mathbb R. \] But then the product contains a sequence with \(x_2=0\) and \(x_3=1\), contradicting the defining relation \(x_2=x_3\). Thus this subset is not a cartesian product of coordinate subsets.