Limits and continuity are the central concepts of calculus in one and several variables. These concepts can be extended to quite general situations; the simplest of these is when there is some way of measuring the distance between two objects. Some of the most important examples of these `metric spaces' occur as sets of functions, so this course looks at ways in which one might say that a sequence of functions converges. Taking these ideas one step further, we look at convergence which does not come from a generalized distance function. These are the ideas of point set topology. The course will include topics such as countability, continuity, uniform convergence and compactness, as well as an introduction to the core areas of function analysis. This will include the notions of Banach and Hilbert spaces, including Reproducing Kernel Hilbert Spaces which are important in Applied Mathematics, Statistics and elsewhere.