The cup and the plate share many of the same mathematical properties; we know this precisely because they are topologically the same. This means that when you

The cup and the plate share many of the same mathematical properties; we know this precisely because they are topologically the same. This means that when you prove that something is (topologically) true for the cup, you know that it is also true for the plate, and for anything "in between". Instead of a statement only applying for the cup, it now applies to a large class of objects, which is very powerful.