Science News has a great description of Topology in their article Tied Up in Knots, which is the field of study my good friend Ken has decided to use as his focus for his academic research (he has a doctorate in the field). His explanations always make sense when he lays it all out, the knowledge tends to slip through my head within a week. So, this article is very helpful, and I’m adding it to my site for future reference. Here’s the heart of the definition:
Topology studies shapes. Specifically, it studies shapes’ properties that are not affected by stretching, moving, twisting, or pulling—anything that doesn’t break up the object or fuse some of its parts. The proverbial example is that, to a topologist, a coffee mug is the same as a doughnut. In your imagination, you can squash the mug into a doughnut shape, and it will retain the property of having a hole, namely its handle.
A sphere is different. You can stretch a sphere into a stick and bend the stick so its ends touch. But turning that open ring into a doughnut will involve fusing the ends, and that’s forbidden.