In Guitarist's manifolds thread a question was asked about the reasons for defining a topology the way we do. I hope to give a bit of light on this.

The trick is to realize that the standard definition of a topology was chosen because it was easiest to work with, not because it was most enlightening. So we have to develop an equivalent way of thinking about what we want to talk about in topology first, abstract that idea properly and than show that we can come to the same set of standard definitions.

So the first question is: What are we trying to study in topology. The answer there is continuity. Now continuity is intricately tied up with the idea of nearness. Recall the first version of the continuity definitions you may have heard. A function on the reals is continuous if nearby points in the domain can't be mapped to far away points in the image. You can extend that to metric spaces with the usual version of continuity but what is important was the idea of nearness. I look at the image and at points that are nearby and than look at the pre-image and points that are nearby and make sure those two sets are related correctly.

Ok, so we know that the idea of nearness is important - lets abstract it. Lets suppose I have a set and on it I want to have some idea of nearness. By that I mean that given any subset of I can tell you what points are near to . I do this by creating a function which tells me which points are near to a subset (namely the points of are near to the points of ).

Now lets see what properties this function should have (ie. what are the abstract ideas of nearness we want to capture?).

The second requirement captures the transitive nature of nearness, if x is near to y and y is near to z than x should be near to z. If there was a point then x would be close to some other point which was close to a point in but not close to any point in , breaking this transitive relationship.

- Well should be near to itself for any reasonable definition of nearness. This implies that
- should find all points that are near to . This means that if we ask what is near to we should not get any new points i.e. .
- The points which are near to subsets and should be the points near to and the points near to i.e.
- Finally, nothing should be close to the empty set. ie.

The last requirement is kind of a symmetry argument, as no point in the set should be important a priori. This implies we have two choices for . If we do not choose it to map to the empty set but want to maintain that no point is important we are forced to map it to (else the points we are either including or excluding have some sort of special status). But for all subsets of we have that that so . To keep things interesting we have to make the first choice,

Now that we have out idea of nearness (a few of you may notice that I have just defined a Kuratowski closure operator), we can give a definition of a continuous function. Recall that we want the image of nearby sets to be nearby. We codify this idea by saying a function from to is continuous if where and are the concepts of nearness on and respectively.

What does this have to do with the normal definition of a topology, I hear you ask? Well, here is the trick. In this formulation, we call a subset closed if and open if its set compliment is closed. These open sets satisfy the normal definition of a topological space. In fact we can reverse the construction and given a family of subsets which satisfies the normal definition of a topology construct its closure operator. Also, families of subsets tend to be easier to work with than operators (especially when trying to build new topological spaces) and so a topological space is typically defined from the view point of open sets and not from closure operators, even though these view points are equivalent.