This is the Introductory lecture to a beginner's course in Algebraic Topology, MATH5665, given by N J Wildberger of the School of Mathematics and Statistics at UNSW in 2010. The course is suitable for 3rd and 4th year mathematics majors, hopefully with some prior knowledge of group theory. Others with some mathematical maturity should be able to follow along with most of the material, which is highly visual.

This first lecture introduces some of the main topics of the course and presents three problems to get students thinking topologically. Topics include: curves, winding number and curvature, two dimensional topological spaces and classification of surfaces using Conway's ZIP proof, polyhedra and Euler number, vector fields, the fundamental group, three dimensional manifolds and quaternions, and homology.

I suggest what might be the two most important and interesting objects in the history of mathematics (you can see if you agree!) I also give three problems. One is a paper cutting exercise, another is Sam Loyd's pencil trick, and the third is a popular puzzle involving a block of wood, a loop of string and two balls.

By studying how Bob would view a dilation in Rachel's framework, we are led to the notion of a generalized dilation. Going from one basis to the other involves a 'Change of basis matrix'.

We show how these ideas lead naturally to the important concepts of eigenvectors and associated eigenvalues, and give examples of finding them.

We continue discussing 2x2 matrices, their interpretation as

Finally we discuss the connections with calculus, introducing the idea

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What is the fundamental problem? It

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We discuss two different ways of drawing 3D objects in 2D, emphasizing