  Magnetism and spin-orbit coupling (SOC) are two quintessential ingredients
underlying novel topological transport phenomena in itinerant ferromagnets.
When spin-polarized bands support nodal points/lines with band degeneracy that
can be lifted by SOC, the nodal structures become a source of Berry curvature;
this leads to a large anomalous Hall effect (AHE). Contrary to
three-dimensional systems that naturally host nodal points/lines,
two-dimensional (2D) systems can possess stable nodal structures only when
proper crystalline symmetry exists. Here we show that 2D spin-polarized band
structures of perovskite oxides generally support symmetry-protected nodal
lines and points that govern both the sign and the magnitude of the AHE. To
demonstrate this, we performed angle-resolved photoemission studies of
ultrathin films of SrRuO$_3$, a representative metallic ferromagnet with SOC.
We show that the sign-changing AHE upon variation in the film thickness,
magnetization, and chemical potential can be well explained by theoretical
models. Our study is the first to directly characterize the topological band
structure of 2D spin-polarized bands and the corresponding AHE, which could
facilitate new switchable devices based on ferromagnetic ultrathin films.
