Topology is a field of math that focuses on how molding or stretching a shape alters the space it takes up.

In a way, topology is similar to geometry. Both fields study shapes. However, geometry focuses on rigid measurements, such as the length of lines in a triangle. It answers questions like: If I change the length of this line, how will the other lines change? In geometry, the new triangle is considered a different triangle altogether.

But topology does not focus on such rigid measurements. Instead, topology treats the study of shapes like a game. As in a game, a topologist sets out certain rules. Then, they explore what shapes are possible within those rules.

For instance, a rule might be that all the shapes in a certain family can morph into one another without tearing or gluing parts together. You could explore this yourself by squeezing a water balloon. As you watch the balloon warp, you see continuous changes to its shape. Squeezing the middle causes the outsides to bulge. But the balloon doesn’t break or form new connections. So, all the shapes the balloon takes still obey the rule.

A mathematician might say that all the shapes of this balloon belong to the same topological group...