Property of Baire

Okay kiddo, let me explain property of Baire in simple terms. Suppose you have a big jar filled with marbles in different colors. You want to put some marbles in a different jar but you want to make sure that all the marbles of one color stay together in both jars. So you decide to take out one color of marbles at a time and put them in the second jar.

Property of Baire is like this process, but instead of marbles, we're talking about sets of numbers. Imagine you have a line that contains all the real numbers and you want to divide that line into two parts - let's call them A and B. You want to make sure that there are no intervals where part of A and part of B overlap.

So, like in the example with the marbles, you start by taking out one interval of numbers at a time and putting them in one of the sets. You keep doing this until you've put all intervals in either A or B. What you end up with is two sets A and B that don't have any overlapping intervals.

This is the property of Baire - you can divide a space into two parts without any overlap if you keep taking out one interval at a time and putting in one of the sets.

I hope that makes sense, let me know if you have any questions!