This course provides a brief introduction to three foundational areas of modern mathematics: set theory, measure theory, and topology. In set theory, we will see how to count well past infinity (ordinal and cardinal arithmetic), and we will also see how set theory forms a logical foundation for the whole of modern mathematics. In topology, we will see how continuous deformation is defined and used (for example in fixed point theorems), and in particular we will look at the concept of the dimension of a set (for example, what makes a line one dimension, or a plane two dimensions). Measure theory asks how one can define and find the ‘content’ of a set, that is ‘how much stuff is in it’ (for example, length of a curve or area of a region); we will see examples of sets, called fractal, for which the most natural measures of how much ‘stuff’ is in them involve thinking of them as sets of fractional dimension.
This course is ideal as an immediate sequel to MAT2115, Introduction to Pure Mathematics. It will be a valuable foundation for anyone considering seriously studying more mathematics; in particular, it will be a prerequisite for Real Analysis in Spring 2014.