Non-Borel set
Okay kiddo, imagine you have a bunch of toys and you want to group them based on their color. You can easily do that with toys that are red, blue or green, right? But what if you have a toy that changes its color every time you look at it? You won't be able to group it with the other toys based on color because its color can be anything at any time!
Now, imagine that instead of toys, we have a bunch of points on a line. We want to group them based on where they are located on the line. We can easily group them if they are in certain sections of the line, like between 0 and 1 or between 2 and 3. But what if we have some points that are in a weird section of the line that doesn't fit into any of these groups? These points are called non-Borel sets.
So basically, we can think of a non-Borel set as a group of points on a line that doesn't fit into any nice little section we can easily name. It's like trying to group toys based on color when some of the toys keep changing color all the time!
Now, imagine that instead of toys, we have a bunch of points on a line. We want to group them based on where they are located on the line. We can easily group them if they are in certain sections of the line, like between 0 and 1 or between 2 and 3. But what if we have some points that are in a weird section of the line that doesn't fit into any of these groups? These points are called non-Borel sets.
So basically, we can think of a non-Borel set as a group of points on a line that doesn't fit into any nice little section we can easily name. It's like trying to group toys based on color when some of the toys keep changing color all the time!