Bounded set (topological vector space)

Okay kiddo, imagine you have a big toy box filled with all your toys. But you can only play with certain toys - let's say your parents put a boundary around a smaller set of toys that are safe for you to play with. This is kind of like a bounded set in math.

In math, a bounded set is a group of points in a space that can be contained within a certain "boundary." This means that no matter how far you try to stretch or move the set, it won't ever go outside of that boundary.

A topological vector space is a fancy math term for a type of space where you can add and subtract points using some special rules. If you have a set of points in a topological vector space, it can be called a bounded set if there is some number (let's call it "r") that can be used as a boundary - this means that all of the points in the set are within some distance of each other, and none of them are too far away from each other.

So, just like your parents watched over your toy box to make sure you only played with safe toys, mathematicians use bounded sets to describe sets of points that stay within a certain distance of each other.