Definition 2

Any subset of $\mathbf{R}^n$ which is the product of n open (resp. closed) intervals of $\mathbf{R}$ is called an open (resp. closed) box in $\mathbf{R}^n$. [For $n = 2$ it is called an open (resp. closed) rectangle.]

The open boxes in $\mathbf{R}^n$ form a base of the topology of $\mathbf{R}^n$ (Chapter I, § 4, no. 1); the open boxes which contain a point $x = (x_i)_{1 \leq i \leq n}$ of $\mathbf{R}^n$ form a fundamental system of neighbourhoods of $x$, and so do the closed boxes of $\mathbf{R}^n$ for which $x$ is an interior point.

Every non-empty open box in $\mathbf{R}^n$ is homeomorphic to $\mathbf{R}^n$ (Chapter IV, § 4, no. 1, Proposition 1).

It follows that, when $n \geq 1$, every non-empty open set in $\mathbf{R}^n$ has the power of the continuum.

An open (resp. closed) cube of $\mathbf{R}^n$ is an open (resp. closed) box which is the product of $n$ bounded intervals of equal length [for $n = 2$, it is called an open (resp. closed) square]; the common length of these intervals is called the side (or side-length) of the cube. The open cubes

form a countable fundamental system of neighbourhoods of the point $x = (x_i)$, as $m$ runs through the set of all integers $> 0$ or through any sequence of integers increasing to infinity.

Every open (or closed) box in $\mathbf{R}^n$ is connected (Chapter I, § 11, no. 4, Proposition 8); in particular, $\mathbf{R}^n$ is connected and locally connected.

If $A$ is a non-empty open set in $\mathbf{R}^n$, its components are therefore open sets (Chapter I, § 11, no. 6, Proposition 11); and the set of these components is countable, for $\mathbf{R}^n$ has a countable dense subset (for example $\mathbf{Q}^n$).

Consider under what conditions a subset $A$ of $\mathbf{R}^n$ will be relatively compact. By Tychonoff’s theorem (Chapter I, § 9, no. 5, Theorem 3) it is necessary and sufficient that the projections of $A$ on the factors of $\mathbf{R}^n$ should be relatively compact; by the Borel-Lebesgue theorem (Chapter IV, § 2, no. 2, Theorem 2) this is equivalent to saying that these projections are bounded subsets of $\mathbf{R}$. We say that a subset $A$ of $\mathbf{R}^n$ is bounded if all its projections are bounded subsets of $\mathbf{R}$; thus we have proved: