Let $X$ be an arbitrary set, are the following statements equivalent?
$$X^{\omega} = \prod_{n \in \mathbb{Z_+}} X$$
$$X^{\omega} = \left\{(x_i)_{i \in \mathbb{Z_+}} \ | \ x_i \in X \right\}$$
i.e. is the following true :
$$X^{\omega} = \prod_{n \in \mathbb{Z_+}} X = \left\{(x_i)_{i \in \mathbb{Z_+}} \ | \ x_i \in X \right\}$$
Is this a non-standard way of representing $X^{\omega}$ in set-builder notation? Are there better/cleaner more standard ways of representing $X^{\omega}$ in set builder notation?