Originally Posted by

**gruff**
Well no, my answer to your question is correct. You didn't ask the right question.

Maybe you're right about that. What I wanted you to see is:

a) containers are not a good way of thinking about sets, as you say;

b) 2 sets A and B are the same if every element in A is in B and every element in B is in A. If A and B are both empty, they are the same, therefore there is only one empty set;

c) there are no special rules for the empty set, the same apply whether whether empty or not.

OK, I'm going to say a bit about open and closed sets. Remember the interval (a, b) on the real line? This interval contains all points between a and b

*but not* a and b. It is called an open interval. It is also a perfectly good open set. Conversely the interval [a, b]

*does* contain a and b, it is a closed interval/set.

You can think of a and b as the boundary points of these intervals, which leads to the conclusion that a closed set contains its boundary, an open set does not.

Here's neat trick. Consider some set A completely embedded in a set B, A subset B. Those elements of B not in A are referred to as the

*complement* of A and written Ac (actually the c should be superscripted, but I can't be arsed).

Now there is a boundary where A "ends" and Ac "begins", but only one. Then the boundary must either belong to A or to Ac, so if A is open, Ac is closed and v. v. (I'm being a bit loose with terminology here, but that needn't worry you for now).

So, the boundary of A can be called bdA. Some more definitions, so take it slow;

the

*interior* of a set A is defined as the largest open set wholly contained in A; if A is open intA = A.

The

*closure* of A, clA, is the smallest closed set wholly containing A. If A is closed clA = A. We can now give a more precise definition of the boundary.

bdA = clA - intA. There is an alternative but equivalenet definition of bdA which we can see later, if you want. Meanwhile, any questions?