Open sets in a metric space (M,d), with examples, Real Analysis II

In this lecture, I introduce the concept of open sets in a general metric space. We start by defining the epsilon neighborhood and discuss how it forms the basis for understanding open sets. I explain the notation we'll use: B_epsilon(x). Using definitions and illustrative diagrams, we explore what it means for a set to be open and how this applies in various metric spaces, including Euclidean space and the discrete metric. We also prove that epsilon neighborhoods are open sets. (MA 426 Real Analysis II, Lecture 5) #RealAnalysis #MetricSpaces #OpenSets #MathLecture #AdvancedMathematics #MathTutorial #Mathematics #DiscreteMetric #advancedcalculus