So, let's talk about point set topology. First, a point set is, as you might have gathered, a set with no algebraic structure (well, let's say - I don't think that is the most general definition).

OK. Let be a set, and let denote the powerset on . Then one says that is atopology oniff the following are true:

finite intersections of elements in (sets, recall) are in ;

arbitrary union of elements in are in ;

.

The "indivisible" pair is called atopological space.

Before continuing, let me hammer this home. Whereas elements in are points, elements in aresets of points.

We shall that this implies that every element in is likewise a set.

Shortly we will allow this bit of terminological abuse: it is customary to refer to a topological space simply as, say (when is is, of course): one says "let be a topological space", the existence of and the underlying set being assumed. but we shan't doing that just yet.

OK? So the elements in (sets, recall), are called theopen setsin . This may seem a bit weird first time around, but I will explain!

Recall we talked about the complement of a set. The closed sets in are those elements in which are the complement in of some set in . Jane would no doubt prefer to say that the closed sets in are elements in the set , and I think on this occasion I would have to agree

Example: Let and suppose that . These are the open sets in

The closed sets (complements in ) are .

From which I hope you can deduce that a set may open, closed, both or neither. Obviously, are both open and closed in any topology.

Which reminds me; I really ought to give some examples of topologies and topological spaces.

Right. Recall that is thesetof real numbers Recall also that this set admits of an order. One says that the "standard" (or usual) topology on is given by , where these are the open sets in .

One calls this the topological real line (or real line for short), and interprets this as the union of the open intervals .

Now recall that the union ofanyarbitrary number of elements in is in . So, . Then and one calls this a closed set in

Does this make sense, or do I need a bucket of cold water throwing over me?