Definition:Z^Z Space

Definition

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

Let $X = \ds \struct {\prod_{i \mathop \in \Z_{\ge 0} } Z, \tau}$ 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 $Z^Z$ (topological) space.

Thus $\tau$ can be referred to as the $Z^Z$ topology.

Also see

• $Z^Z$ Topology is Topology

• Results about the $Z^Z$ space can be found here.

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.): Part $\text {II}$: Counterexamples: $102$. $Z^Z$