Example

\* The product $\mathbf{R}^n$ of $n$ spaces identical with the real line $\mathbf{R}$ is called real number space of $n$ dimensions; $\mathbf{R}^2$ is also called the real plane (cf. Chapter VI, § 1). Likewise, starting from the rational line $\mathbf{Q}$, we define the rational number space of $n$ dimensions $\mathbf{Q}^n$ (rational plane for $n = 2$).

The topology of the space $\mathbf{R}^n$ has as a base the set of all products of $n$ open intervals in $\mathbf{R}$, which are called open boxes of $n$ dimensions. The open boxes which contain a point $x \in \mathbf{R}^n$ form a fundamental system of neighbourhoods of this point. Likewise the products of $n$ closed intervals in $\mathbf{R}$ are called closed boxes of $n$ dimensions. The closed boxes to which $x$ is interior also form a fundamental system of neighbourhoods of $x$. There are analogous results for $\mathbf{Q}^n$. \*