Brouwer's Theorem on the Invariance of Domain

The purpose of this thesis is to present some dimension theory of separable metric spaces, and with the theory developed, prove Brouwer’s Theorem on the Invariance of Domain. This theorem states, that if we embed a subset of the n-dimensional Euclidean space into the aforementioned space, this embedding is an open map. We begin by revising some elementary theory of point-set topology, that should be familiar to any graduate student in mathematics. Drawing from these rudiments, we move on to the concept of dimension. The dimension theory presented is based on the notion of the small inductive dimension. We define this dimension function for regular spaces and state and prove various results that hold for this function. Although this dimension function is defined on regular spaces, we mainly focus on separable metric spaces. Among other things, we prove that the small inductive dimension of the Euclidean n-space is exactly n. This proof makes use of the famous Brouwer Fixed-Point Theorem, which we naturally also prove. We give a combinatorial proof of the Fixed-Point Theorem, which relies on Sperner’s lemma. We move on to develop some theory regarding the extensions of functions. These various results on extensions allow us to finally prove the theorem that lent its name to this thesis: Brouwer’s Theorem on the Invariance of Domain