Topological techniques have been applied to data analysis and visualization in a wide range of use cases, spanning combustion, biology, neuroscience, cosmology, and material science. However, these techniques face scalability challenges. For example, topological algorithms assume that building a global structure is necessary, reducing their parallel scalability. Moreover, topological data structures often impose a large memory overhead compared to the data size, usually a factor of 10 or more, which requires more resources to perform the analysis. We propose a new foundation for scalar field topology that attempts to address these scalability challenges. We formalize an abstraction of topological queries that creates a well-defined interface between topological analysis programs and query algorithms. This interface is not tied to a specific data structure. These topological queries give users a set of primitives that can be combined in interesting ways to develop custom analyses. Moreover, queries grant topology researchers freedom to invent new data structures and algorithms because they no longer need to compute a specific data structure such as a merge tree or Morse-Smale complex. We use this new flexibility to design localized topological data structures that exhibit linear scaling of construction at a small cost of query performance in most practical cases.