Almost open set

Well sweetie, let’s think about a room in our house. We can say the room is ‘open’ if we can walk inside it and not bump into anything, or if we can see everything inside it from the doorway. But what if there’s a step up or a small object on the floor that we could trip over? We might still call the room ‘almost open’ because it’s mostly easy to walk through or see inside, but there are a few places we need to be careful of.

This is just like an ‘almost open’ set in math. Imagine we have a big space that we call a set, and we put some points inside it. We say the set is ‘open’ if we can draw a circle around any point in the set without going outside the set. But sometimes there are points on the edge of the set where we can draw a circle, but if we go slightly outside that circle we’d be outside the set. That means the set isn’t completely ‘open’, but it’s ‘almost open’ because we can get very close to the edge without leaving the set.

So an ‘almost open set’ is like a room with a few obstacles we need to watch out for, or a set with some points close to the edge. We can still get inside or stay inside, but we need to be a bit careful.