3. Real projective spaces

In this section we shall need to invoke constantly the notions and results on quotient spaces (Chapter I, § 3), and in particular the following two properties, which for convenience we state as lemmas:

Let $E$ be a topological space, $R$ an equivalence relation on $E$, $A$ a subset of $E$, $R_A$ the equivalence relation induced on $A$ by $R$ and let $f$ be the canonical mapping of $E$ onto $E/R$. Then:

1. Topology of real projective spaces

We recall the following definitions from algebra: given a division ring or a field K, the set of lines passing through o (i.e. vector subspaces of dimension 1) in the left vector space K_s^{n+1} over K is called left projective space of n dimensions over K, and is denoted by P_n(K).

If we make correspond to every line passing through o in K_s^{n+1} the same line with the origin omitted, we have a bijection of P_n(K) onto the quotient of K_{n+1}^* (the complement of {o} in K_s^{n+1}) by the following equivalence relation Δ_n(K) between vectors x and y of K_{n+1}^*: "there exists t ∈ K such that t ≠ o and y = tx". In what follows we shall identify P_n(K) with this quotient set. In the theory of projective spaces, we take the interval [0, n] of N as index set for the coordinates of a point of K_{n+1}^*. The coordinates x_i (0 ≤ i ≤ n) of any one of the points of K_{n+1}^* whose canonical image is x ∈ P_n(K) constitute what is called a system of homogeneous coordinates of the point x.

For each integer p such that −1 ≤ p ≤ n, the canonical image in P_n(K) of a vector subspace of p + 1 dimensions (without the origin) of K_s^{n+1} is called a projective linear variety of p dimensions. A system of p points of P_n(K) is said to be free if it consists of the canonical images of p points of K_{n+1}^* which form a free system in the vector space K_s^{n+1}. The projective linear variety of P_n(K) generated by a free system of p + 1 points (i.e. the smallest projective linear variety which contains these p + 1 points) has dimension p.

When K is the field R, the corresponding projective spaces can be topologized, and it is these we shall study.

The projective space P_1(R) is called the real projective line, and P_2(R) is called the real projective plane.

Whenever there is no risk of confusion, we shall write P_n and Δ_n instead of P_n(R) and Δ_n(R).

2. Projective linear varieties

Every injective linear mapping $f$ of $\mathbf{R}^{n+1}$ into $\mathbf{R}^{m+1}$ ($m \geq n$) defines, by restriction to $\mathbf{R}_{n+1}^*$ and then by passage to the quotient with respect to the relations $\Delta_n$ and $\Delta_m$ (Set Theory, R, $\S$ 5, no 8), an injective mapping g of $P_n$ into $P_m$, called a projective linear mapping. If $\varphi$ (resp. $\psi$) is the canonical mapping of $\mathbf{R}_{n+1}^*$ onto $P_n$ (resp. $\mathbf{R}_{m+1}^*$ onto $P_m$), we have $g \circ \varphi = \psi \circ f$, which shows that g is continuous on $P_n$ (Chapter I, $\S$ 3, no. 4, Corollary to Proposition 6). In particular, every projective linear transformation of $P_n$ (i.e. projective linear mapping of $P_n$ onto itself) is a homeomorphism of $P_n$ onto itself.

We recall also that, if V and V' are two projective linear varieties of p dimensions in $P_n$, there exists a projective linear transformation of $P_n$ which transforms V into V'. In particular, if $p \geq 0$, there exists a projective linear transformation which transforms V into a coordinate projective linear variety, that is to say the canonical image of a coordinate variety W' of $p + 1$ dimensions (without the point o) of $\mathbf{R}^{n+1}$. If we identify W' with $\mathbf{R}_{p+1}^*$, the relation induced by $\Delta_n$ on W' is precisely $\Delta_p$; since W' is closed and saturated with respect to $\Delta_n$, Lemma 1 shows that V' is homeomorphic to $P_p$ and is closed in $P_n$; moreover, if $p < n$, its complement is dense in $P_n$ ($\S$ 1, no. 4, Corollary to Proposition 3). Hence:

3. Embedding real number space in projective space

Proposition 5 of no. 2 shows that if we are given a projective hyperplane $\mathbf{H}$ in $\mathbf{P}_n$ ($n \geq 0$), there is a homeomorphism of $\mathbf{R}^n$ onto the complement $\mathbf{C}H$ of this hyperplane. Once $\mathbf{H}$ has been chosen it is often convenient to identify $\mathbf{R}^n$ and $\mathbf{C}H$ by means of the homeomorphism defined in Proposition 5; the projective hyperplane $H$ is then said to be "at infinity", and so are its points and subsets. Usually one takes $H$ to be the "coordinate" hyperplane whose equation is $x_0 = 0$; and then the point $z = (z_1)$ of $\mathbf{R}^n$ is identified with the point of $P_n$ whose homogeneous coordinates are $1, z_1, z_2, \ldots, z_n$.

Once this identification has been made, the closure in $P_n$ of any affine linear variety $V$ of $p$ dimensions in $\mathbf{R}^n$ is a projective linear variety of $p$ dimensions, not contained in the hyperplane at infinity, and identical with the projective linear variety generated by $V$. Conversely, every projective linear variety $P$ of $p$ dimensions which is not contained in the hyperplane at infinity has as its trace on $\mathbf{R}^n$ an affine linear variety of $p$ dimensions, whose closure in $P_n$ is $P$.

In the particular case $n = 1$, the hyperplane at infinity is a point; since $P_1$ is compact, it follows from Alexandroff's theorem (Chapter I, § 9, no. 8, Theorem 4) that $P_1$ is homeomorphic to the space $\tilde{\mathbf{R}}$ obtained by compactifying the locally compact space $\mathbf{R}$ by the adjunction of a point (the "point at infinity"). By Proposition 4 of § 2, no. 4, we see also that the real projective line $P_1' \mathbf{R}$ is homeomorphic to the circle $S_1$ and to the torus $T$.

On the other hand, if $n > 1$, $P_n(\mathbf{R})$ is not homeomorphic to $S_n$, as we shall see later (cf. Exercise 4).

The "point at infinity" of the space $\tilde{\mathbf{R}}$ is denoted by $\infty$, with no sign attached. It is important not to confuse $\tilde{\mathbf{R}}$ with the extended real line $\mathbf{R}$ defined in Chapter IV, § 4, which has two "points at infinity"; indeed $\tilde{\mathbf{R}}$ is homeomorphic to the quotient space of $\mathbf{R}$ obtained by identifying the two points $+\infty$ and $-\infty$.

4. Application to the extension of real-valued functions

Since $\mathbf{R}$ can be considered as a subset of $\tilde{\mathbf{R}}$, every mapping of a set $E$ into $\mathbf{R}$ (i.e. every real-valued function on $E$) can be considered as a mapping of $E$ into $\tilde{\mathbf{R}}$; in particular, if $E$ is a subset of a topological space $F$ and $f$ is a mapping of $E$ into $\mathbf{R}$, it may happen that at some points of the closure $F_c$ of $E$, $f(x)$ tends to the limit $\infty$ as $x$ tends to one of these points while remaining in $E$; we may then extend the function $f$ by continuity by assigning it the value $\infty$ at these points (Chapter I, § 8, no. 5, Theorem 1).

Consider in particular the case where $E$ is a subset of $\mathbf{R}^n$, the space $\mathbf{R}^n$ itself being considered as embedded in projective space $P_n$; if we suppose that the hyperplane at infinity is $x_0 = 0$, a real-valued function $f$ defined on $E$ may be identified with the mapping

of $E$ into $\tilde{\mathbf{R}}$; from what has been said in the previous paragraph it may be possible to extend this function, not only to some points of $\mathbf{R}^n$ in the closure of $E$, but also to some of the "points at infinity" of $\mathbf{P}_n$ in the closure of $E$.

Let us show that we obtain in this way, for example, the extension by continuity to the whole of $\tilde{\mathbf{R}}$ of a rational function of a real variable, already defined in algebra. Let us identify $\tilde{\mathbf{R}}$ and $\mathbf{P}_1$, every real number $x \in \mathbf{R}$ being identified with the point whose homogeneous coordinates are $(1, x)$. and the point $\infty$ with the point whose homogeneous coordinates are $(0, 1)$. Let $u(x, y)$ be a rational function, $u$ and $v$ being two coprime polynomials of degrees $m$ and $n$ respectively; if we suppose for example that $m \leq n$, and if we put $u_1(x, y) = x^n u(x, y)$, $v_1(x, y) = x^n v(x, y)$, the rational function $u/v$ may be considered as the restriction, to the set of real numbers $x$ such that $v(x) \neq 0$, of the mapping $(x, y) \to (u_1(x, y), v_1(x, y))$. In other words, we extend $u/v$ by continuity, by giving it the value $\infty$ at those points $x \in \mathbf{R}$ where $v(x) = 0$, and by giving it at the point $\infty$ the value $0$ if $m < n$, the value $\infty$ if $m > n$, and the value of the ratio of the leading coefficients if $m = n$.

In particular, the function $1/x$ can be extended to the point $0$ by taking the value $\infty$ there, to $\infty$ by taking the value $0$ there; this extended function is evidently a homeomorphism of $\tilde{\mathbf{R}}$ onto itself. The same is true of the homographic function $(ax + b)/(cx + d)$ when $ad - bc \neq 0$.

Likewise, if $n$ is an integer $> 0$, the function $x^n$ extends to the point $\infty$ by taking the value $\infty$ there.

On the other hand, it is in general impossible to extend by continuity a rational function of two real variables either to the space $\mathbf{P}_1 \times \mathbf{P}_1$ or to the space $\mathbf{P}_2$ (cf. Exercise 5).

5. Spaces of projective linear varieties

Given a division ring $K$, the set $\mathbf{P}_{n,p}(K)$ of projective linear varieties of $p \geq 0$ dimensions of the left projective space $\mathbf{P}_n(K)$ is clearly in one-to-one correspondence with the set of vector subspaces of $p - 1$ dimensions of the left vector space $K_s^{p-1}$. Let $L_{s-1,s-1}(K)$ denote the set of free systems $(x_i)_{1 \leq i \leq s-1}$ of $p - 1$ vectors of $K_s^{p-1}$; then the set $\mathbf{P}_{n,p}(K)$ is again in one-to-one correspondence with the quotient of $L_{n+1,\ p+1}(K)$ by the equivalence relation $\Delta_{n,\ p}(K)$: " $(x^k)$ and $(y^k)$ generate the same vector subspace of $p + 1$ dimensions of $K_s^{n+1}$". In what follows we shall identify $P_{n,\ p}(K)$ with this quotient set. On the other hand, if to each free system $(x_k)$ of $p + 1$ vectors of $K_s^{n+1}$ we make correspond the matrix $X$ of $p + 1$ rows and $n + 1$ columns for which $x_k$ is the kth row ($1 \leq k \leq p + 1$), we have a one-to-one correspondence between $L_{n+1,\ p+1}(K)$ and the set of all matrices of $p + 1$ rows and $n + 1$ columns which are of rank $p + 1$; we shall identify $L_{n+1,\ p+1}(K)$ with this set of matrices, and the relation $\Delta_{n,\ p}(K)$ between two matrices $X, Y$ is then the following: "there exists a non-singular square matrix $T$ of order $p + 1$ such that $Y = T.X$".

In what follows we shall take $K$ to be the field $\mathbf{R}$, and we shall omit the letter $K$ in the notation above. We can define a topology on $P_{n,\ p}$ by a process which generalizes the definition of the topology of real projective spaces. Namely, $L_{n+1,\ p+1}$ is contained in the space $M_{p+1,\ n+1}$ of all matrices of $p + 1$ rows and $n + 1$ columns with real elements; we endow $L_{n+1,\ p+1}$ with the topology induced by the topology of this matrix space ($\S \ 1$, no. 6).

We shall use the following notation: given a matrix $X$ of $p + 1$ rows and $n + 1$ columns, and any strictly increasing sequence

of $p + 1$ indices belonging to the interval $[0, n]$ of $\mathbf{N}$, we denote by $X_\sigma$ the square submatrix of $X$ formed by the columns whose indices are $i_1, i_2, \ldots, i_{p+1}$. We denote by $A_\sigma$ the subset of $L_{p+1,\ p+1}$ consisting of these matrices $X$ such that $X_\sigma$ is non-singular. By Proposition 6 of $\S \ 1$, no. 6, $A_\sigma$ is a dense open set in $M_{p+1,\ n+1}$, and the function $X \to X_\sigma^{-1}$ is continuous on $A_\sigma$.

A geometrical interpretation of the set $A_\sigma$ is as follows: let $E_\sigma$ be the vector subspace of $\mathbf{R}^{n+1}$ generated by the vectors $e_i$ of the canonical basis such that $i \in \sigma$, and let $E'_\sigma$ be the complementary subspace generated by the $e_i$ such that $i \notin \sigma$; then to say that a matrix $x$ belongs to $A_\sigma$ means that the projections on $E_\sigma$ of its $p + 1$ rows of $x_k$ form a free system, or again that the vector subspace generated by the $x_k$ is a complement of $E'_\sigma$ (or that its intersection with $E'_\sigma$ consists only of 0).

6. Grassmannians

If K is a field and X is any matrix of L_{n+1,p+1}(K), let c_σ(X) denote the determinant of X_σ; in this way, to each matrix X of L_{n+1,p+1}(K) correspond

determinants, not all zero (the components of the exterior product of the p + 1 rows of X). If we make correspond to X the point of the projective space P_{n-1}(K) whose homogeneous coordinates are the c_σ(X), we have defined a mapping of L_{n+1,p+1}(K) into P_{n-1}(K), compatible with the relation Δ_{n,p}(K); passing to the quotient, we have therefore a mapping f of P_{n,p}(K) into P_{n-1}(K). The image G_{n,p}(K) of P_{n,p}(K) under this mapping is called the Grassmannian of indices n, p. We recall also that the mapping f is injective, for if X is a matrix such that X_σ is non-singular, the matrix Y = X_σ^{-1}X of B_σ which corresponds to the class of X mod. Δ_{n,p}(K) is the matrix (d_{ij}/c_σ(X)) (1 ≤ i ≤ p + 1, 0 ≤ j ≤ n), where d_{ij} denotes the determinant of the matrix obtained from $X_{\sigma}$ by replacing the $i$th column of $X_{\sigma}$ by the $j$th column of $X$ [which implies that $d_{ij}$ is, up to sign, equal to one of the $c_{\tau}(X)$].

When $K$ is the field $\mathbf{R}$, this mapping $f$ is evidently continuous. The inverse mapping $g$ is also continuous; for the elements of a matrix belonging to $B_{\sigma}$ are rational functions of the homogeneous coordinates of the point of the Grassmannian to which it corresponds; since $f(B_{\sigma}) = B'_{\sigma}$ is the set of points of $G_{n,p}$ whose homogeneous coordinate with index $\sigma$ is not 0, it is an open set in $G_{n,p}$; hence $g$ is continuous at every point of $B'_{\sigma}$, and since every point of $G_{n,p}$ belongs to at least one set $B'_{\sigma}$, $g$ is continuous at every point. Thus:

Exercises

See the exercises for § 3.