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 $Z_{\ge 0}$.

This page needs proofreading. In particular: Completely out of my depth here. S&S do not specify the topology on $\ds \prod_{\alpha \mathop \in A} Z_\alpha$, so I am assuming the Tychonoff topology. It is also unclear whether the notation used $Z^+$ actually includes zero or not. I am assuming that the latter point may not actually matter here, as the ordering and algebraic structure of $\Z$ appears not to be used directly, and that all we want is for $\Z$ to have a well-enough defined lower bound. If you believe all issues are dealt with, please remove {{Proofread}} from the code. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Proofread}} from the code.

Also see

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

• Results about the uncountable product space of $Z_{\ge 0}$ can be found here.

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.): Part $\text {II}$: Counterexamples: $103$. Uncountable Products of $Z^+$