Homotopy theory is a branch of mathematics that studies continuous deformations of functions and spaces. In the context of topological spaces, homotopy theory allows us to understand the fundamental properties and relationships between different topological spaces. It helps in determining when two continuous maps between spaces can be continuously deformed into one another, which provides insight into the underlying structure and connectivity of the spaces. Homotopy theory is particularly useful in classifying topological spaces and studying their properties such as connectedness, compactness, and dimensionality.