Some remarks on density topologies on the plane (Q1308830)

Let \(d\) be the density topology in \(\mathbb{R}\) and let \(d^ 2\) be the ordinary density topology on the plane \(\mathbb{R}^ 2\). Some topological properties (spread, weight, Lindelöf-degree, etc.) of the spaces \((\mathbb{R}^ 2, d\times d)\) and \((\mathbb{R}^ 2, d^ 2)\) are investigated. It is shown that these spaces are not homeomorphic.