Toplogy

Course Overview

This course introduces homeomorphisms, metric spaces, compactifications, manifolds and more!.

As you progress through this course, you will find links to related topics in other courses (such as calculus and partial differential equations). This interconnected structure is designed to help you quickly revisit prerequisite ideas without breaking your workflow.

Introduction to topology

A word that is often thrown around in our everyday language is shape. But, what exactly does it mean to have a shape? Do the number of holes in an object play a role in the shape? Are all holes created equal? Do rulers and sheets of paper have the same shape? In mathematics we must present precise definitions in order to explain exactly what we mean by particular terms, and when discussing shapes (topology), this will remain true.

Definition: A topological space is a set (which represents the entire space) \(X\) together with a collection of subsets \(\mathcal{U} = \{U_\alpha\}_{\alpha \in A}\) which we call open sets. The open sets have the following properties:

1. \(X\) is open.
2. \(\emptyset\) is open.
3. If \(\{U_\beta\}_{\beta \in B}\) are some of the open sets, then \(\mathop{\cup} \limits_{\beta \in B}U_\beta\) is also an open set.
4. If \(U\) and \(V\) are open sets, then so is \(U \cap V\).

Mappings

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More to come!