Suggestion of simple and interesting topics on topology. I am taking a general topology class and since we have just finished our schedule (almost everything from Engelking and something of Baire spaces), our professor said that we could choose any cool topic to give a lecture. It just have to be simple enough so everyone can understand and interesting enough to keep our classmates awake.
Have you got any suggestions?
Thank you!

A few ideas, depending on your personal taste. Some are very difficult but could make a good lecture to give an idea of applications of topology in other sciences.

Sphere eversion (see also Wikipedia). There was an article in (the french edition of) Scientific American, long ago, that made me dream...
But maybe not so simple.

The Banach-Tarski paradox. This requires some measure theory.

Space-filling curves. Can lead to a few nice plots, like the dragon curve.

The Jordan curve theorem. An apparently obvious result. Only apparently.

Fractals and topological dimension. 

Knot theory

Much more difficult (but maybe ok for a quick overview): dynamical systems, KAM theory and applications in celestial mechanics and the stability of the solar system.

Also very difficult: general relativity and spacetime topology. For this one, I'd suggest the book "Gravitation" by Misner, Thorne and Wheeler.
