Metric spaces concerns itself with the notion of distances in abstract sets and leads into the beginnings of Topology. This is by far my favourite module.

The way we measure distances seems like the obvious way to do so, but there is no reason we should do it that way. We would like a distance to convey the notion of how close two objects are. Before we define the notion of abstract distance we look at the properties of distances that are important. 1) A distance needs to be positive and should only be 0 when it is between two points that are the same, if this were not the case then it wouldn’t really make sense. 2) The distance from a point x to a point y should be the same as the distance from y to x. 3) There needs to be a shortest distance between any two objects. That is if we were to travel between points via another point then the distance needs to be either greater than or equal to the distance between our point.
So we now define a distance (or metric) as any function that has these 3 conditions and is well-defined on our space. But why do we need to bother with all this. Firstly there is not always an obvious notion of distance in some spaces, for example if we take the space of all continuous functions (in this space our points become functions), what is the distance between sin(x) and e^(x)? Another reason why it’s important is so that we can define an "open ball," this is the set of all pints in our space that are less than a fixed distance away. These sets will obviously be different depending on which metric w use. Now it gets complicated, if we abstract the notion of openess we get into a whole new field of mathematics, topology. Whereas geometry could be thought of as the study of the properties of spaces which are left unchanged after a transformation, topology can be thought of as the study of the properties of a space which are left unchanged after a deformation. Think of a space as a rubber sheet which can be stretched and compacted as much as you like. So now things like shape, distance and angles become meaningless, a square can be deformed into a circle. What topologists are interested in is what properties are the same after the "stretching" takes place. These properties are things like openess, compactness, connectedness and continuity.

There is an old joke that a topologist doesn’t know the difference between a coffee mug and a doughnut, this is because to them it is the same space, just deformed.

By: stuartpilbrow