Definition:Uncountable Product Space of Positive Integers

Definition
Let $Z = \struct {\Z_{\ge 0}, \tau_d}$ denote the positive integers with the discrete topology.

Let $A$ be an uncountable set with cardinality $\lambda$.

Let $X_\lambda = \ds \struct {\prod_{\alpha \mathop \in A} Z_\alpha, \tau}$ be the space formed on the countable Cartesian product of instances of $Z$ such that $\tau$ is the Tychonoff product topology.

Then $X$ is known as the uncountable product space of $Z_{\ge 0}$.

Thus $\tau$ can be referred to as the uncountable product topology of $Z_{\ge 0}$.

Also see

 * Uncountable Product Topology of $Z_{\ge 0}$ is Topology