# The significance of Topologically Equivalence (To A Donut)?

A friend of mine explained how a coffee mug is topologically equivalent to a donut to me tonight. I have to say the idea is very interesting! However, I don't know topology a lot but I am wondering about the significance of such equivalence. Not to mention that my friend failed to give a satisfying answer. So my question really is: Why is topologically equivalence important? How can such a equivalence help us? Under what circumstances? What problem are we trying to solve? I'd apologize if this is not quite a good question to ask. But I'd appreciate if someone who knows topology well can share some insight in this!

• The original motivation of topological equivalence is not usefulness, but pure curiosity: when are two figures the same shape? What does it mean for them to be the same shape? This is the answer. One of the fun parts of mathematics is taking an intuitive concept that even lay people have and formalizing a rigorous, working definition of it. (This doesn't mean the shape of a space isn't important or doesn't occur in applications though.) – anon Dec 23 '13 at 6:06
• This may be of some interest, though you may not appreciate it fully if you aren't already familiar with topology. – Michael Albanese Dec 23 '13 at 6:30
• Thanks for the link, the physics part is hard for me to read but I see how topology is used to as a way to describe a certain state. I might not regard that as an application of topology theory, but as ronno pointed out, it can be regard as general property, which might be useful under some circumstances. – user48601 Dec 23 '13 at 6:51