Defining and exemplifying the different types of connectedness in topology

Abstract

Connectedness is a highly important topological concept, present in various branches of mathematics. This qualitative and deductive study aims to define and relate, in the most general way possible, three different types of connectedness in topology: connectedness, path connectedness, and local connectedness. It also presents properties of arbitrary topological spaces that satisfy these definitions, as well as the development of all the necessary theory for understanding these concepts, including the notions of open sets, closed sets, bases, topological subspaces, interior, closure, and boundary of a set. It is proven that connectedness is preserved under continuous functions, which provides a way to verify when two sets are not homeomorphic. It is also concluded that path connectedness is a stronger form of connectedness, as it is shown that every path-connected set is also connected. Furthermore, this work contributes to the understanding of abstract topological concepts by providing examples involving subsets of ℝⁿ.