Definition 2

The quotient space $P_{n,p}(\mathbf{C})$ of the topological space $L_{n+1,\ p+1}(\mathbf{C})$ by the equivalence relation $\Delta_{n,p}(\mathbf{C})$ is called the space of projective linear varieties of $p \geqslant 0$ dimensions in the projective space $P_n(\mathbf{C})$.

By the argument of Proposition 6 of Chapter VI, § 3, no. 5, we see first of all that $P_{n,p}(\mathbf{C})$ is Hausdorff. Next we prove that it is compact by replacing the subspace $V_{n+1,\ p+1}$ (in the proof of Proposition 7 of Chapter VI, § 3, no. 5) by the subspace $W_{n+1,\ p+1}$ of $L_{n+1,\ p+1}(\mathbf{C})$ consisting of systems of $p + 1$ vectors which form an orthonormal Hermitian basis of the vector subspace they generate; that is to say, $W_{n+1,\ p+1}$ consists of the matrices $X = (x_{ij})$ which satisfy the conditions

The proof of Proposition 8 of Chapter VI, § 3, no. 5 extends unaltered for the space $P_{n,p}(\mathbf{C})$ and shows that this space is connected and locally connected and that each of its points has a neighbourhood homeomorphic to $\mathbf{C}^{p+1,(n-p)}$. Finally the proof of Proposition 9 of Chapter VI, § 3, no. 6 applies without change, and therefore the Grassmannian $G_{n,p}(\mathbf{C})$ is homeomorphic to $\mathbf{P}_{n,p}(\mathbf{C})$.