Differential geometry and topology

Differential geometry and topology are two different types of math. Differential geometry is all about finding the shapes of curved things, like the shape of a sphere or the surface of a mountain. Topology is all about studying shapes that can bend and stretch in different ways. So if you think of a rubber band, you can easily move it around, twist it, and stretch it, but it always stays the same shape. Topology tries to figure out how different shapes can move around and change without changing their overall shape. It's like trying to figure out all the different ways a rubber band can move without ever changing its shape.