As promised. First recall I and others have said that a manifold is a topological space with certain additional properties, and also that a topological space space is a set with certain additional properties. So it seems wise to start with some exceptionally basic set theory.

In what follows, I will cover no more ground than is needed in order to move on. And, oh; if you think you know this stuff already (and I'm sure you do!) it still might be worth a quick skim, as not everyone uses the same notational and terminological conventions.

So a set is simply a collection of objects that has some claim to being a collection. If this sounds circular, it is because the notion of a "set" is in some sense primitive - we all know what we mean, just that we can't quite put our finger on it. I write to mean that is anelementin the set and its negation .

Context sometimes make it more convenient to write to mean exactly the same thing - is a set that contains the element , likewise the negation.

Now notice this key fact: elements in a set can beabsolutely anything-they may, and often are, themselves sets, as we shall see. So in order to avoid a famous paradox due to B. Russell, I make the stipulation.....

A set is well-defined iff every object in the universe of objects is either definitely in that set or definitely not in it.

Key fact 2: Given a set and another set I will say that if every element in is also an element in I will say say that is a subset of . But note I do not NECESSARILY insist there are elements in that are NOT also in .....but I might.

In the latter case I write , in the former I write Note this does not mean I am uncertain whether or not , rather that whatever I have to say about this subset applies equally whether equality holds or whether it doesn't.

Let me know introduce you some extremely important subsets. Suppose a set. Then the set is called the "singleton set", since it has only a single element. We shall see a generalization of this shortly, but note for now that , since is always true, is meaningless.

The other subset of key importance is the empty set which I write as for any set . In a while I shall prove the rather startling fact that the empty set is always a subset of any set.

This post is already over-long, but let me just mention 2 "operations" on sets (of course they are not really operations)

The intersection of 2 sets defines those elements that are in both sets equally. So for the sets I write . Note that the intersection is itself a set.

The union of 2 sets defines those elements that are in one, the other, or both. I write for the union, which is itself a set.

Some properties of intersection and union will be of interest, but I have no doubt you have had enough of me for now.

Ask questions if you want....