Topology is the study of the fundamental structure of spaces, we disregard distance, throw geometry out of the window and exist only in a deformable world where up is down and breathing space is all that matters. This breathing space we characterise by way of open sets- subsets of our world where we can "move around" given that we are a sufficiently small person (though size matters not) of given dimension.

We thus identify spaces whose open set structure is the same, a definition conveniently attended to by a certain kind of "nice" function called a homeomorphism. Those spaces with one of these homeomorphism type bad boys between them are the same in a very subtle way, and naturally those that differ often do so in the same subtle way- and it is our job as mathematicians to point to these subtle differences with enormous algebra shaped arrows.

This is where invariants come in: an invariant is something which, when fed through a homeomorphism, comes out the same: just as cardinality is preserved by bijections, so our invariants are preserved by homeomorphisms. Different invariant thus implies different topology- and this provides fuel for those classifications theorems we all so love.

We would like, in an ideal world, to just ascribe a number to each topology and be done with it, just like our cardinals. Sorted. Classified.

But some things are more subtle than that, and more numerous- and each topology gives a whole rich world of new and unusual algebraic forms... just waiting to be discovered.

MS03. 7:30- followed by a general retirement to the grad for beverage based relief.

The dream is that this will be the first in a series of lectures, building to the derivation of a particularly powerful and sexy invariant- the alexander polynomial of knots.