In Topology, When Are Two Shapes the Same?

Kevin Hartnett in Quanta:

Topologists study the properties of general versions of shapes, called manifolds. Their animating goal is to classify them. In that effort, there are a few key distinctions. What exactly are manifolds, and what notion of sameness do we have in mind when we compare them?

Here are the basic differences.

Manifolds can be shapes of any dimension, from zero-dimensional points to one-dimensional lines to two-dimensional surfaces (like the surface of a ball) to 100-dimensional spaces (and beyond) that are hard to picture but as mathematically real as anything else. Mathematicians study them because, among other reasons, three- and four-dimensional manifolds provide the setting of our lives.

More here.