TITLE: How to think of a set?
QUESTION [16 upvotes]: I am doing self study for the last two months on functional analysis. As I get a bit used to the terms like space, topology, manifold, etc, etc, I realized that everything is defined in terms of set. If I look at the wikipedia page on set or set theory we can get quite a bit of explanation, but it did not clear me how to think of a 'set'. May be I am trying to think 'set' as some geometric object from which everything an be derived. May be I am wondering how a physicist would think of 'set'. I am worried that this question will be closed with lot of down vote but I still decided to try my luck. May be I asking few remarks from community how they think of 'set'. That might help me to develop some missing link, which I cannot explain properly.

REPLY [3 votes]: A set can be thought of as an inclusion rule, or "All the objects which satisfy a given inclusion rule."
"All odd numbers".  If you're an odd number, you're in the set.  If you're not, you're not in the set.  Or, "All the states in the US whose names start with 'A'".  
The set might be empty: "All odd multiples of 2".  The empty set is actually important, but you'll learn that as you go.  Just like zero is an extremely important and useful number but they don't teach you that at first.
You can even have a set defined like this: "The numbers 1, 2, and 5".  Anything that is not one of those numbers is not in that set.