Nature evolves structures like honeycombs at optimized performance with limited material. These efficient structures can be artificially created with the collaboration of structural topology optimization and additive manufacturing. However, the extensive computation cost of topology optimization causes low mesh resolution, long solving time, and rough boundaries that fail to match the requirements for meeting the growing personal fabrication demands and printing capability. Therefore, we propose the neural synthesizing topology optimization that leverages a self-supervised coordinate-based network to optimize structures with significantly shorter computation time, where the network encodes the structural material layout as an implicit function of coordinates. Continuous solution space is further generated from optimization tasks under varying boundary conditions or constraints for users' instant inference of novel solutions. We demonstrate the system's efficacy for a broad usage scenario through numerical experiments and 3D printing.