Exercise 6, § 4

Let $K$ be a non-discrete Hausdorff topological division ring (or field).

a) Endow the left projective space $P_n(K)$ with the quotient of the topology of $K_{n+1}^*$ by the equivalence relation $\Delta_n(K)$. Extend Propositions 1, 4 and 5 of Chapter VI, § 3, to $P_n(K)$ endowed with this topology. Show that $P_n(K)$ is connected if $K$ is connected, and otherwise is totally disconnected.

\* b) If $K$ is locally compact (but not discrete), show that $P_n(K)$ is compact. (Let $U$ be a compact neighbourhood of $0$ in $K$; let $a \in K$ be such that

let $S$ be the subset of $K_{n+1}^*$ consisting of points $x = (x_i)$, all of whose coordinates $x_i$ belong to $U$, and such that $x_k \in a \cdot \overline{U}$ for some index $k$; show that $P_n(K)$ is the canonical image of $S$.)

c) Likewise, extend Definition 2 and Propositions 6 and 8 of Chapter VI, § 3 to the set $P_{n,p}(K)$ of projective linear varieties of $p$ dimensions in $P_n(K)$; \* also Proposition 7, assuming that $K$ is locally compact [for Proposition 7 consider, for each sequence $\sigma$ of indices, the subset $S_\sigma$ of $A_\sigma$ consisting of matrices all of whose rows belong to the set $S$ defined in b); show that $P_{n,p}(K)$ is the canonical image of the union of the sets $S_\sigma$]. \* Generalize Proposition 9 of Chapter VI, § 3, when $K$ is a field.

d) If $K$ is not commutative, let $P_{n,p}'(K)$ denote the space of $p$-dimensional projective linear varieties in the right projective space of $n$ dimensions over K [the topology of $P'_{n,p}(K)$ being defined in the same way as that of $P_{n,p}(K)$]. Show that $P_{n,p}(K)$ and $P'_{n,n-p-1}(K)$ are homeomorphic [to every $(p+1)$-dimensional vector subspace $V$ of the left vector space $E = K_s^{n+1}$ make correspond the orthogonal complement of $V$, which is a vector subspace of the dual $E^*$ of $E$].