On topological projective spaces and their Grassmannians
In an n-dimensional Desarguesian topological projective space \({\mathbb{P}}\) the sets \({\mathcal T}_ j\) of j-flats carry natural topologies \(\tau_ j\) for \(0\leq j<n\). The author shows that the topological spaces (\({\mathcal T}_ j,\tau_ j)\) are either all connected or all totally disconnected. Analogously, these spaces are either all compact or all non locally compact. It is known that the topologies \(\tau_ j\) are uniquely determined by \(\tau_ 0\). In the sequel of the paper the author assumes that only the sets \({\mathcal T}_ h\) and \({\mathcal T}_{h-1}\) are endowed with topologies for some \(0<h<n\). He investigates conditions which assure that these topologies turn \({\mathbb{P}}\) into a topological projective space with special emphasis on the case of locally compact and connected topologies.
Desarguesian topological projective space
scientific article; zbMATH DE number 4178011
scientific article