When I was in undergrad, I was endlessly enamored with topology. I loved how it took intuitive ideas like “wiggle room” and “nearness”, and turned them into precise math. At the heart of the subject is one notion: the open set. So in this issue, I’ll explain open sets with pictures! Armed with that, we’ll explain the formal definition of a topological space. an open set looks like a potato Here’s what I want you to visualize when I say “open set”: This is a blob with the insides filled in, but not including the boundary. defining this precisely A subset U of R2 is open if for every point p in U, there is an open ball B containing p (of some radius) such that B is contained in U. Intuitively, as p moves closer and closer to the boundary of the set, the open ball B containing it would have to get smaller and smaller. this is not an open set This set (containing its boundary) not be open since for the point on the boundary, no open ball containing it would be contained inside the set.