[proofplan] We prove discreteness by showing that every subset of the finite product is open. The key point is that, in a discrete factor, every singleton $\{x_i\}$ is open, and in a finite product the product of finitely many open subsets is a basic [open set](/page/Open%20Set). Hence every singleton in $X_1 \times \cdots \times X_n$ is open, and an arbitrary subset is a union of such open singletons. [/proofplan] [step:Show that every singleton in the product is open] Let $P := X_1 \times \cdots \times X_n$ denote the product set, and let $\tau$ denote the [product topology](/page/Product%20Topology) on $P$. Fix an arbitrary point $x \in P$. Write $x = (x_1,\dots,x_n)$, where $x_i \in X_i$ for each $i \in \{1,\dots,n\}$. Since $(X_i,\tau_i)$ is discrete, every subset of $X_i$ is open in $\tau_i$; in particular, $\{x_i\} \in \tau_i$. By the definition of the finite product topology, the set \begin{align*} U_x := \{x_1\} \times \cdots \times \{x_n\} \end{align*} is a basic open subset of $P$, hence $U_x \in \tau$. By the definition of the Cartesian product, \begin{align*} U_x = \{(x_1,\dots,x_n)\} = \{x\}. \end{align*} Thus $\{x\}$ is open in $P$. [guided] Let $P := X_1 \times \cdots \times X_n$ be the underlying set of the product, and let $\tau$ be the product topology on $P$. To prove that $P$ is discrete, it is enough to first understand the smallest possible subsets, namely singletons. Choose an arbitrary point $x \in P$. Since $P$ is the Cartesian product, there are uniquely determined points $x_i \in X_i$ for $i \in \{1,\dots,n\}$ such that $x = (x_1,\dots,x_n)$. For each factor, the hypothesis says that $(X_i,\tau_i)$ is discrete. By definition of the [discrete topology](/page/Discrete%20Topology), every subset of $X_i$ is open in $\tau_i$. Applying this to the subset $\{x_i\} \subset X_i$, we get $\{x_i\} \in \tau_i$ for every $i$. Now use the definition of the finite product topology. In a finite product, a product of open subsets of the factors is a basic open set. Therefore \begin{align*} U_x := \{x_1\} \times \cdots \times \{x_n\} \end{align*} is a basic open subset of $P$, and hence $U_x \in \tau$. Finally, this basic open set contains exactly one point. Indeed, a point $y \in P$ lies in $U_x$ precisely when $y = (y_1,\dots,y_n)$ with $y_i \in \{x_i\}$ for every $i$. This forces $y_i = x_i$ for every $i$, so $y = x$. Thus \begin{align*} U_x = \{x\}. \end{align*} Therefore the singleton $\{x\}$ is open in the product topology. [/guided] [/step] [step:Conclude that every subset of the product is open] Let $A \subset P$ be arbitrary. Since every singleton $\{x\}$ with $x \in P$ is open in $\tau$, and since topologies are closed under arbitrary unions, the set \begin{align*} A = \bigcup_{x \in A} \{x\} \end{align*} is open in $\tau$. If $A = \varnothing$, this is the empty union, and $\varnothing \in \tau$ by the axioms of a topology. Hence every subset of $P$ is open, so $\tau$ is the discrete topology on $P$. Therefore $X_1 \times \cdots \times X_n$, equipped with the product topology, is discrete. [/step]