As anyone who fought their way through analysis in their first term will know: very little in maths comes easy. The most fundamental and fundamentally obvious properties of spaces can be hidden in a world of axiomatic obfuscation, leaving you wondering whether such an obvious truth should be a fair assumption to make. More often than not the answer is "No!" "...and go sit in the corner 'til you learn to be a proper mathematician!"- these dubious assumptions need to be proved from first principles, and it is not likely to be easy.

Of all the techniques used to classify topological spaces, few are so simple, so fruitful and so ubiquitous as homology and its dual, cohomology. By breaking a space into discretized parts we spot patterns, monitoring the closed n-dimensional "loops" to give groups, smaller structures that elucidate the whole; yielding vital theorems that, with a little case analysis, can classify the almost indeterminably abstract.