We study two-dimensional hexagonal photonic lattices of silicon Mie resonators with a topological optical band structure in the visible spectral range. We use 30 keV electrons focused to nanoscale spots to map the local optical density of states in topological photonic lattices with deeply subwavelength resolution. By slightly shrinking or expanding the unit cell, we form hexagonal superstructures and observe the opening of a band gap and a splitting of the double-degenerate Dirac cones, which correspond to topologically trivial and nontrivial phases. Optical transmission spectroscopy shows evidence of topological edge states at the domain walls between topological and trivial lattices.