Dense subset

Imagine you have a bunch of toys that you want to keep in a box. Some toys are bigger and take up more space than others. Now, if you want to make sure you can fit as many toys as possible in the box, you could try to put the smaller toys in the gaps left by the bigger toys. This way, you would have a lot of toys in the box even though some of the toys take up more space than others.

This is sort of like what it means for a set to be dense. In math, sets are kind of like groups of numbers or objects. A dense subset is a smaller set that occupies as much space as possible inside a larger set. Just like the smaller toys fill in the gaps left by the bigger toys in the toy box, a dense subset fills in all the spaces left by the larger set.

This might all sound a bit abstract, so let's use a number line as an example. Imagine you have a number line that goes from 0 to 1. There are an infinite number of numbers between 0 and 1 – you could have 0.1, 0.01, 0.001, and so on. But what if you just had the numbers 0.1 and 0.3? These two numbers are not very many, but they do a good job of filling up the space between 0 and 1. In this case, the set {0.1, 0.3} is said to be a dense subset of the larger set from 0 to 1.

So, to sum up, a dense subset is a smaller group of numbers or objects that fills in all the spaces left by a larger group. It's like putting the smaller toys in the gaps left by the bigger toys in a toy box.