In this lecture we use the abstract machinery developed last time to prove the higher-dimensional generalisation of the celebrated Jordan Curve Theorem.

Most six-year olds would have no difficulty in establishing the validity of the Jordan Curve Theorem: namely, that if you draw a circle on a piece of paper then there is an "inside" and an "outside". However it is possible to confuse the poor child by making the circle look less like a circle and more like a maze.

Some beautiful pictures that illustrate this concept can be found in the article The Jordan Curve Theorem is non-trivial by Fiona Ross and William T. Ross, available online here.

The higher-dimensional version says that if $f \colon S^{n-1} \to S^n$ is an embedding then $S^n \setminus f(S^{n-1})$ has two components. This is called the Jordan-Brouwer Separation Theorem and is much less suitable for entertaining kids with.