In the first week, we introduce the concept of a topological space and homeomorphism as the natural notion of equivalence. We will see ways in which to construct topological spaces, such as subspaces, products and quotients. We will introduce the Fundamental Problem and the idea of topological invariants. In addition, we will get to know a range of important examples of topological spaces such as the sphere and the Möbius strip, that will serve as recurring examples throughout this course.

**Learning outcomes**

After this week, you should be able to

- construct topological spaces using products, disjoint unions, and quotients;
- list diverseĀ
**examples**of topological spaces; - explain the idea of a homeomorphism, topological invariants and their use.

**Tasks and Materials**

- Have a look at the lecture notes when they appear.
- Browse through the introductory pages of Hatcher’s book.
- Have a look at the example sheet published this week. The problems should be handed in on Thursday of Week 2!

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