We present a method for inferring dense depth maps from images and sparse depth measurements by leveraging synthetic data to learn the association of sparse point clouds with dense natural shapes, and using the image as evidence to validate the predicted depth map. Our learned prior for natural shapes uses only sparse depth as input, not images, so the method is not affected by the covariate shift when attempting to transfer learned models from synthetic data to real ones. This allows us to use abundant synthetic data with ground truth to learn the most difficult component of the reconstruction process, which is topology estimation, and use the image to refine the prediction based on photometric evidence. Our approach uses fewer parameters than previous methods, yet, achieves the state of the art on both indoor and outdoor benchmark datasets. Code available at: https://github.com/alexklwong/learning-topology-synthetic-data.