On algebraic and topological properties of continuous functions over a compact Hausdorff space

Abstract

For a compact Hausdor topological space, we shall consider a space of all continuous real-valued functions. The central problem of this dissertation is to translate algebraic properties of this collection of functions into topological properties of the underlaying topological space. This involves considering the function space as a ring. One of the major facts is that this function ring is completely determined by the underlaying topological space. This is a result by Leonard Gillman and Meyer Jerison. Therefore we need to specify conditions under which, conversely the topological space can be determined by the algebraic structure of the function space. The topological aspect requires us to specify the collection of subsets that allow us to consider the function space as a topological space with the compact-open topology. The fact that we have enough points to work with is facilitated by Urysohn's Lemma. A more critical investigation that leads us to dense subsets of the function space is developed and facilitated by Stone-Weierstrasss theorem. We narrow down our discussion by considering a unit closed interval as a compact space.