CHAPTER 3 - Topological Spaces

Abstract

A topological space is a set of points, along with a set of neighbourhoods for each point, that satisfy a set of axioms relating points and neighbourhoods. The definition of a topological space relies only upon set theory and is the most general notion of a mathematical space that allows for the definition of concepts, such as continuity, connectedness, and convergence. Other spaces such as manifolds and metric spaces are the specializations of topological spaces with extra structures or constraints. The topological spaces are a central unifying notion and appear in virtually every branch of modern mathematics. The branch of mathematics that studies topological spaces in their own right is called point-set topology or general topology. The concept of a topological space is introduced in terms of the axioms for the open sets. The closure operator is just one of a number of operators that can be used to define a topology. To completely determine a topological space, both the points and the open sets in the space should be specified. The two topological spaces are the same if both the points and the family of open sets are the same in each.