Homotopy Theory is a branch of mathematics that studies the properties of topological spaces that are preserved under continuous transformations. It focuses on the concept of homotopy, which is a way to deform one continuous function into another without breaking or tearing. This theory helps mathematicians understand the shape and structure of spaces by examining how they can be transformed into each other. In Homotopy Theory, two spaces are considered equivalent if they can be continuously transformed into one another, leading to the idea of homotopy equivalence. This concept is fundamental in areas such as algebraic topology, where tools like homotopy groups and homology are used to classify spaces. The work of mathematicians like Henri Poincaré laid the groundwork for this field, influencing many modern developments.