Proposition 2

Every $p$-dimensional linear affine variety in $\mathbf{R}^n$ is a closed subset of $\mathbf{R}^n$, homeomorphic to $\mathbf{R}^p$.

It is this result which is the origin of the names line and plane given to affine linear varieties of one and two dimensions in a vector space over an arbitrary division ring. We recall also that, for $n \geq 1$, the affine linear varieties of $n - 1$ dimensions of $\mathbf{R}^n$ are called hyperplanes (loc. cit.).

The $n$ one-dimensional coordinate varieties, that is to say the $n$ lines passing through $0$ and the $n$ points $e_i$ respectively, are called the coordinate axes. For $n = 2$ the axis through $e_1$ is sometimes called the axis of abscissas and the axis through $e_2$ is called the axis of ordinates; the first coordinate of a point $x \in \mathbf{R}^2$ is called its abscissa, the second coordinate its ordinate.

Every line $D$ passing through a point $a$ has a parametric representation $t \to a + tb$, where $t$ runs through $\mathbf{R}$ and $b \neq 0$; this mapping is a homeomorphism of $\mathbf{R}$ onto $D$. The vector $b$ is called a direction vector of D, and its components $b_i (1 \leq i \leq n)$ are called direction ratios of D. If $b'$ is another direction vector of D, there exists $h \neq 0$ in $\mathbf{R}$ such that $b' = hb$.

The set of points $a + tb$, where $t$ runs through the set of real numbers $\geq 0$, is called the closed ray (or simply ray, or half-line) with origin $a$ and direction vector $b$ (or with direction ratios $b_i$). It is a closed subset of $\mathbf{R}^n$, homeomorphic to the interval $[0, +\infty[$ of $\mathbf{R}$, and therefore connected. The line D is the union of the two rays with origin $a$ and direction vectors $b$ and $-b$ respectively; these rays are said to be opposite.

By abuse of language, the set of points $a + tb$, where $t$ runs through the set of real numbers $> 0$, is called the open ray with origin $a$ and direction vector $b$; it is homeomorphic to the interval ]$0, +\infty[$ (and therefore homeomorphic to $\mathbf{R}$), but is not open in $\mathbf{R}^n$ if $n > 1$, although it is open in the line which contains it.

A line passing through two distinct points $x$ and $y$ also has a parametric representation $(u, v) \to ux + vy$, where $(u, v)$ runs through the set of pairs of real numbers such that $u + v = 1$. Given any two points $x, y$ (distinct or not) the set of points $ux + uy$, where $(u, v)$ runs through the set of pairs of real numbers such that $u \geq 0, v \geq 0$ and $u + v = 1$, is called the closed segment (or simply segment) with end-points $x, y$. A closed segment is compact and connected, for if its end-points are distinct it is homeomorphic to the interval $[0, 1]$ of $\mathbf{R}$.

If $x \neq y$, the set of points $ux + vy$ such that $u > 0, v > 0$ and $u + v = 1$ is called (by abuse of language) the open segment with end-points $x, y$; it is homeomorphic to the open interval ]$0, 1[$ (and hence also homeomorphic to $\mathbf{R}$). Finally the union of $\{y\}$ and the open segment with end-points $x, y$ is sometimes called the segment open at $x$ and closed at $y$; it is homeomorphic to the interval $[0, 1[$. All the segments with $x$ and $y$ as end-points are connected, and the closure of each of them is the closed segment with the same end-points.