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The central idea of metamaterials and metaoptics is that, besides their base materials, the geometry of structures offers a broad extra dimension to explore for exotic functionalities. Here, we discover that the topology of structures fundamentally dictates the topological properties of optical near fields and offers a new dimension to exploit for optical functionalities that are irrelevant to specific material constituents or structural geometries. We find that the nontrivial topology of metal structures ensures the birth of polarization singularities (PSs) in the near field with rich morphologies and intriguing spatial evolutions including merging, bifurcation, and topological transition. By mapping the PSs to non-Hermitian exceptional points and employing homotopy theory, we extract the core invariant that governs the topological classification of the PSs and the conservation law that regulates their spatial evolutions. The results have effectively bridged three vibrant fields of singular optics, topological photonics, and non-Hermitian physics, with potential applications in chiral sensing, chiral quantum optics, and beyond photonics in other wave systems.