ThmDex – An index of mathematical definitions, results, and conjectures.

Countable cartesian product with empty set is empty

Formulation 0

Let $\emptyset$ be the D13: Empty set.
Let $X_n$ be a D11: Set for each $n \in \mathbb{N}$ such that

(i)	\begin{equation} \exists \, n \in \mathbb{N} : X_n = \emptyset \end{equation}

Then \begin{equation} \prod_{n = 0}^{\infty} X_n = \emptyset \end{equation}

Proofs

Proof 0

Let $\emptyset$ be the D13: Empty set.
Let $X_n$ be a D11: Set for each $n \in \mathbb{N}$ such that

(i)	\begin{equation} \exists \, n \in \mathbb{N} : X_n = \emptyset \end{equation}

This result is a particular case of R4264: Cartesian product with empty set is empty. $\square$