1. Real number space $\mathbf{R}^n$

1. The topology of $\mathbf{R}^n$

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:

2. The additive group $\mathbf{R}^n$

The set $\mathbf{R}^n$, endowed with the group structure which is the product of the additive group structures of the $n$ factors of $\mathbf{R}^n$, is an abelian group; we use the additive notation, the sum of $x = (x_i)$ and $y = (y_i)$ being therefore $x + y = (x_i + y_i)$. The topology of the number space is compatible with this group structure; endowed with these two structures, $\mathbf{R}^n$ is a topological group called the additive group of n-dimensional real number space. If $n = 0$, we make the convention that $\mathbf{R}^0$ denotes a group consisting only of the identity element.

The uniform structure of this group, called the additive uniformity of $\mathbf{R}^n$, is the product of the uniformities of the factors of $\mathbf{R}^n$ (Chapter III, § 3, no. 2). If, for each integer $p > 0$, $V_p$ denotes the set of pairs $(x, y)$ of $\mathbf{R}^n$ such that $\max_{1 \leq i \leq n} |x_i - y_i| \leq 1/p$, the sets $V_p$ form a fundamental system of entourages of this uniformity. Whenever we consider $\mathbf{R}^n$ as a uniform space we shall always have in mind the additive uniformity just defined, unless the contrary is expressly stated. Endowed with this uniform structure, $\mathbf{R}^n$ is a complete uniform space (Chapter II, , no. 5, Proposition 10)

3. The vector space $\mathbf{R}^n$

Since $\mathbf{R}$ is a field, we can define on $\mathbf{R}^n$ a vector space structure over the field $\mathbf{R}$, the product $t x$ of a scalar $t \in \mathbf{R}$ and a point (or vector) $x = (x_i)$ of $\mathbf{R}^n$ being the point $(t x_i)$. Note that the homothety $(t, x) \to t x$ is continuous on $\mathbf{R} \times \mathbf{R}^n$. If $e_i$ denotes the vector of $\mathbf{R}^n$ all of whose coordinates are zero, except for that of index $i$, which is equal to 1, then the $e_i$ form a basis of the vector space $\mathbf{R}^n$, called the canonical basis of this space. Every vector $x = (x_i) \in \mathbf{R}^n$ can be written as $x = \sum_{i=1}^n x_i e_i$, and the relation $\sum_{i=1}^n t_i e_i = 0$ implies that $t_i = 0$ for $1 \leq i \leq n$.

The vector space $\mathbf{R}^n$ is therefore of dimension $n$ over the field $\mathbf{R}$, in the sense of algebra (Algebra, Chapter II, § 7, no. 2); hence its name of $n$-dimensional real number space.

Let $f$ be an affine mapping of the vector space $\mathbf{R}^n$ into the vector space $\mathbf{R}^m$ ($m$ and $n$ being integers > 0). If we put $g(x) = f(x) - f(0)$, $g$ is a linear mapping of $\mathbf{R}^n$ into $\mathbf{R}^m$. Let $a_{ij}$ ($1 \leq j \leq m$) be the coordinates of $g(e_i)$ in $\mathbf{R}^m$ and let $b_j$ ($1 \leq j \leq m$) be those of $f(0)$; if $x_i$ ($1 \leq i \leq n$) is the $i$th coordinate of $x \in \mathbf{R}^n$ and if $y_j$ is the $j$th coordinate of $y = f(x)$, we have

Since every linear polynomial in $x_1, x_2, \ldots, x_n$ is uniformly continuous on $\mathbf{R}^n$, it follows that every affine mapping of $\mathbf{R}^n$ into $\mathbf{R}^m$ is uniformly continuous on $\mathbf{R}^n$ (Chapter II, § 2, no. 6, Proposition 7).

In particular, we know that every affine mapping of $\mathbf{R}^n$ onto itself is bijective and that its inverse is again an affine mapping; hence every affine mapping of $\mathbf{R}^n$ onto itself is a homeomorphism (and an automorphism of the uniform structure of $\mathbf{R}^n$).

Let $(a_i)_{1 \leq i \leq n}$ be a free system of $n$ vectors of $\mathbf{R}^n$ [in other words a basis of the vector space $\mathbf{R}^n$]; if $b$ is any point of $\mathbf{R}^n$, the set P of points $x = b + \sum_{i=1}^{n} u_i a_i$ such that $-1 \leq u_i \leq 1$ for $1 \leq i \leq n$ is a compact neighbourhood of $b$; for there exists a bijective affine mapping $f$ of $\mathbf{R}^n$ onto itself such that $f(b) = 0, f(b + a_i) = e_i$ for $1 \leq i \leq n$; and $f(P)$ is the cube which is the product of the $n$ intervals $[ -1, +1 ]$ in the $n$ factor spaces. $P$ is called the closed parallelootope with centre $b$ and basis vectors $a_i$. The interior of $P$ consists of the points $b + \sum_{i=1}^{n} u_i a_i$ such that $-1 < u_i < 1$ for $1 \leq i \leq n$; it is called the open parallelootope with centre $b$ and basis vectors $a_i$.

4. Affine linear varieties in $\mathbf{R}^n$

Given a $p$-dimensional affine linear variety $V$ in $\mathbf{R}^n$, there exists an affine mapping $f$ of $\mathbf{R}^n$ onto itself which transforms $V$ into a $p$-dimensional coordinate variety, that is to say a vector subspace $V'$ generated by $p$ of the vectors of the canonical basis $(e_i)$ of $\mathbf{R}^n$. There exists a mapping of $V'$ onto $\mathbf{R}^p$ which is an isomorphism of the vector space structure and the topology of $V'$ onto the corresponding structures of $\mathbf{R}^p$ (it is moreover often convenient to identify $\mathbf{R}^p$ with such a coordinate variety $V'$, e.g., with the vector subspace generated by $e_1, e_2, \ldots, e_p$). In addition, $V'$ is a closed subset of $\mathbf{R}^n$ (Chapter I, § 4, no. 3, Corollary to Proposition 7); hence:

5. Topology of vector spaces and algebras over the field $\mathbf{R}$

Let $E$ be a vector space of dimension $n$ over the field $\mathbf{R}$; if $(a_i)_{1 \leq i \leq n}$ is a basis of $E$, then every point $x \in E$ can be written uniquely in the form $x = \sum_{i=1}^n x_i a_i$, where the $x_i$ are real numbers; the mapping $(x_i) \to \sum_{i=1}^n x_i a_i$ is therefore a bijective linear mapping of $\mathbf{R}^n$ onto $E$.

If we transport to $E$ the topology of $\mathbf{R}^n$ by this mapping, $E$ is endowed with a topology compatible with its additive group structure, and the mapping $(t, x) \to tx$ of $\mathbf{R} \times E$ into $E$ is continuous with respect to this topology. This topology is independent of the basis chosen in $E$; for if $(a')$ is another basis of $E$, and if $x = \sum_{i=1}^n x'_i a'_i = \sum_{i=1}^n x_i a_i$, the mapping $(x_i) \to (x'_i)$ of $\mathbf{R}^n$ onto itself is linear and therefore a homeomorphism.

This fact leads one to suspect that the topology so defined on $E$ should be capable of characterization without the help of a basis of $E$. In fact, we shall see later that this is the only Hausdorff topology on $E$ for which the functions $x - y$ (on $E \times E$) and $tx$ (on $\mathbf{R} \times E$) are continuous.

If now $A$ is an algebra of finite rank $n$ over the field $\mathbf{R}$, the above topology on $A$ (considered as an $n$-dimensional vector space over $\mathbf{R}$) is compatible not only with the additive group structure of $A$, but also with its ring structure. This is a consequence of the following more general result:

6. Topology of matrix spaces over $\mathbf{R}$

An important example of a vector space over $\mathbf{R}$ is the space $\mathbf{M}_{m,n}(\mathbf{R})$ of matrices with m rows and n columns whose elements belong to $\mathbf{R}$; this is a space of dimension $mn$ over $\mathbf{R}$, hence is homeomorphic to $\mathbf{R}^{mn}$. By Proposition 5 of § 5, the product $X.Y$ of two matrices $X \in \mathbf{M}_{m,n}(\mathbf{R}), Y \in \mathbf{M}_{n,p}(\mathbf{R})$ is a continuous function of $(X, Y)$. In particular, the topology of the space $\mathbf{M}_n(\mathbf{R})$ of square matrices of order $n$ is compatible with the ring structure on $\mathbf{M}_n(\mathbf{R})$. Furthermore:

Exercises

See the exercises for § 1.