Projective limits of topological spaces I: oddities

This month, let me investigate projective limits of topological spaces.  That is an area of mathematics that is fraught with pitfalls, and I will describe a number of odd situations that can occur in that domain.  You will have to wait until next month (sorry!) to learn about a very nice result on projective limits of compact sober spaces, due to O. Gabber.  Read the full post.