Topological properties are characteristics of a space that remain unchanged under continuous transformations, such as stretching or bending, but not tearing or gluing. These properties help classify shapes and spaces in mathematics, particularly in the field of topology. Common examples include connectedness, compactness, and continuity. In topology, two objects are considered equivalent if one can be transformed into the other without cutting or gluing. This means that a coffee cup and a donut are topologically equivalent because each has one hole. Understanding these properties is essential for studying complex structures in various scientific fields.