but what a topologist can do (..) is identify and use properties these shapes have in common.

The "shapes" we will study are called Topological Spaces.
These properties are  open, closed, connected, compact, having different types of "holes".
We will study each of these properties, and also, transformations between these shapes that preserve some of these properties.
We will also discuss  applications of topology to the "real world", for instance, what is the shape of our universe?