Proposition 8

$P_{n,p}$ is connected and locally connected, and every point has an open neighbourhood homeomorphic to $\mathbf{R}^{(p+1)(n-p)}$.

For every (strictly increasing) sequence of indices $\sigma$, the set $A_\sigma$ is open in $L_{n+1,p+1}$ and saturated with respect to $\Delta_{n,p}$; its canonical image

C_σ in P_{n,p} is therefore an open set homeomorphic to the quotient of A_σ by the equivalence relation Θ_σ induced on A_σ by Δ_{n,p} (Lemma 1).

Let B_σ be the subset of A_σ consisting of matrices X such that X_σ is the unit matrix of order p + 1; the elements of X other than those of X_σ are then arbitrary, and therefore B_σ is homeomorphic to the space R^{(p+1)(n-p)}. To each matrix X ∈ A_σ let us make correspond the matrix Y = X_σ^{-1}X, which belongs to B_σ; then we have defined a continuous mapping g of A_σ onto B_σ, such that g(X) is the only matrix of B_σ congruent to X mod Θ_σ. It follows that B_σ is homeomorphic to A_σ/Θ_σ (Lemma 2), hence to C_σ.

The set C_σ is therefore connected. Since A_σ is dense in L_{n+1,p+1}, C_σ is dense in P_{n,p} and therefore P_{n,p} is also connected (Chapter I, § 11, no. 1, Proposition 1). On the other hand, every point of P_{n,p} belongs to C_σ for at least one sequence of indices σ, and therefore has an open neighbourhood homeomorphic to R^{(p+1)(n-p)}.

The matrix Y = g(X) can be interpreted as follows: let us suppose for the sake of simplicity that the sequence σ consists of the p + 1 indices n − p, n − p + 1, ..., n, and let a_{ij} (1 ≤ i ≤ p + 1, 0 ≤ j ≤ n − p − 1) denote the elements of the first n − p columns of y; then the vector subspace of R^{n+1} generated by the rows of x is that defined by the equations