Topology: General Topology 2021

Let \(n \in \NN\text{.}\) Recall that \(\PP_{\RR}^n\) is the set of 1-dimensional \(\RR\)-linear subspaces of \(\RR^{n+1}\text{;}\) we call these subspaces (real) lines. Define a map

that carries a nonzero vector \(x = (x_0,\dots,x_n) \in \RR^{n+1}\setminus \{0\}\) to the line \([x] = [x_0:\dots:x_n] \in \PP_{\RR}^n\) it spans. Endow \(\PP_{\RR}^n\) with the finest topology such that \(q\) is continuous. Equivalently, \(\PP_{\RR}^n\) is the quotient space \(\left(\RR^{n+1}\setminus \{0\}\right)/\sim\text{,}\) where we declare \(x \sim y\) if and only if there exists a nonzero real number \(\lambda\) such that \(x = \lambda y\text{.}\) Let \(\PP^n_{\CC}\) denote the set of 1-dimensional \(\CC\)-linear subspaces of \(\CC^{n+1}\text{;}\) we call these subspaces complex lines. These subspaces have 1 complex dimension but two real dimensions.) Define a map

that carries any nonzero vector \(z = (z_0,\dots,z_n) \in \CC^{n+1}\setminus \{0\}\) to the line \([z] = [z_0,\dots,z_n] \in \PP^n_{\CC}\) it spans. Endow \(\PP^n\) with the finest topology such that \(q\) is continuous. Equivalently, \(\PP^n_{\CC}\) is the quotient space \(\left(\CC^{n+1}\setminus \{0\}\right)/\sim\text{,}\) where we declare \(w \sim z\) if and only if there exists a nonzero complex number \(\lambda\) such that \(w = \lambda z\text{.}\)