While it is easy to “see” the topology on the point set of the real affine plane, this is not so for the line set. The same phenomenon occurs for topological projective planes and spaces.
The authors succeed in improving this situation for the projective \(n\)- space \(\mathbb{P}_ n(K)=:\mathbb{P}\) over a topological skew-field \(K\): The vector space \(V= K^{n+1}\) is endowed with the product topology, and subsequently the set \(V^ k\). Its subset \(E_ k\) of independent \(k\)- tuples of vectors inherits the subspace topology. Denote by \(\sigma_ k:E_ k \to_ k \mathbb{P}:=\) set of \(k\)-dimensional subspaces, the map assigning to an independent \(k\)-tuple the subspace generated by its vectors. Now \(_ k\mathbb{P}\) is endowed with the quotient topology with respect to \(\sigma\), and \(\mathbb{P}\) with the sum topology (free union) of the \(_ k\mathbb{P}\). The authors give a short proof that then \(\mathbb{P}\) is a topological projective space and its topology coincides with the one constructed inductively in the pioneering paper of J. Misfeld [Abh. Math. Semin. Univ. Hamb. 32, 232-263 (1968; Zbl 0164.208)].
In addition to this theorem (2.10) they show: \(\mathbb{P}\) satisfies the separation axiom \(T_ 3\). Discreteness, second countability and local compactness are hereditary properties, i.e. satisfied either by all or by none of the spaces \(K\), \(_ 0\mathbb{P}\), \(_ 1\mathbb{P},\ldots,_{n-1}\mathbb{P}\). They also give a short proof for a result by C. Zanella [Abh. Math. Semin. Univ. Hamb. 59, 125-142 (1989; Zbl 0715.51006)]: If \(K\) is locally compact and not discrete, then \(\mathbb{P}\) is compact and has a countable basis.