Curvilinear structure segmentation (CSS) is under semantic segmentation, whose applications include crack detection, aerial road extraction, and biomedical image segmentation. In general, geometric topology and pixel-wise features are two critical aspects of CSS. However, most semantic segmentation methods only focus on enhancing feature representations while existing CSS techniques emphasize preserving topology alone. In this paper, we present a Joint Topology-preserving and Feature-refinement Network (JTFN) that jointly models global topology and refined features based on an iterative feedback learning strategy. Specifically, we explore the structure of objects to help preserve corresponding topologies of predicted masks, thus design a reciprocative two-stream module for CSS and boundary detection. In addition, we introduce such topology-aware predictions as feedback guidance that refines attentive features by supplementing and enhancing saliencies. To the best of our knowledge, this is the first work that jointly addresses topology preserving and feature refinement for CSS. We evaluate JTFN on four datasets of diverse applications: Crack500, CrackTree200, Roads, and DRIVE. Results show that JTFN performs best in comparison with alternative methods. Code is available.1