<a href="https://www.youtube.com/watch?v=tMw9W1aXDNE&feature=youtu.be">Video</a> of the lecture.<br>
<a href="https://www.youtube.com/watch?v=kq_T0Yq-oVQ">Waiting room video</a>: interview with Dusa McDuff<br><br>

Started by emphasizing that, in our "proof" that the Euler characteristic of the 2-sphere S<sup>2</sup> is V-E+F=2, we used the fact that a simple closed curve on the 2-sphere cuts the 2-sphere into two pieces. Then observed that a simple closed curve on a m&ouml;bius strip or on a torus does not necessarily cut the space into two pieces. So we should really justify (i.e. prove) that a simple closed curve on the 2-sphere always cuts it into two pieces.<br><br>

Gave some motivation for a more precise approach to topology and stated the Jordan Curve Theorem. Explained that the next few lectures would focus on obtaining a good understanding of the statement of this theorem and on the main concepts in its proof. Eventually you'll be able to read page 112 of Armstrong's book for more details on the Jordan Curve Theorem -- but there are some definitions and tools you need to acquire first. 
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Ended by giving the definition of a topological space (Definition 2.1 on page 28 of Armstrong's book). Next lecture we'll consider more examples aimed at reinforcing our understanding of this definition.