Point-set topology is a branch of mathematics that studies the properties of space that are preserved under continuous transformations. It focuses on concepts such as open and closed sets, convergence, and compactness, which help define the structure of different types of spaces. In point-set topology, a set of points is analyzed in terms of its neighborhood and limit points. This framework allows mathematicians to explore various types of spaces, including metric spaces, topological spaces, and Hausdorff spaces, providing a foundation for more advanced topics in topology and analysis.