A gas of magnons in magnetic films differs from all other known systems demonstrating Bose-Einstein condensation (BEC), since it possesses two energetically degenerate lowest-energy quantum states with non-zero wave vectors ±k BEC. Therefore, BEC in this system results in a spontaneously formed two-component Bose-Einstein condensate described by a linear combination of two spatially non-uniform wave-functions ∝ exp(±ik BECz), while condensates found in other physical systems are characterized by spatially uniform wave-functions. Here we report a study of BEC of magnons with sub-micrometer spatial resolution. We experimentally confirm the existence of the two wave-functions and show that their interference results in a non-uniform ground state of the condensate with the density oscillating in space. Additionally, we observe stable topological defects in the condensate. By comparing the experimental results with predictions of a theoretical model based on the Ginzburg-Landau equation, we identify these defects as quantized vortices.