R is a connected space! | Intro to Topology INTUITION + RIGOR!

The video is an introduction to point-set topology and the notion of connected spaces from the basics, in the concrete example of the set of real numbers. In the beginning, I explain what are connected spaces, open sets etc. in very simple visual examples in the plane, showing both intuitive and rigorous definitions of these concepts. I prove that R is a connected topological space and explain the intuition behind the proof very visually, before formalizing the proof using analytical concepts such as the completeness axiom/least upper bound property in R. The video is designed for people familiar with the basics of set theory (unions and intersection of sets), and ideally who are learning concepts in real analysis. The goal is to give a clear overview of a major aspect of topology together with practicing intuition and writing rigorous mathematical proofs.

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