Topology Puzzles

Posted by John Baez

Let’s say the closed unit interval [ 0 , 1 ] [0,1] maps onto a metric space X X if there is a continuous map from [ 0 , 1 ] [0,1] onto X X . Similarly for the Cantor set.

Puzzle 0. Does the Cantor set map onto the closed unit interval, and/or vice versa?

Puzzle 1. Which metric spaces does the closed unit interval map onto?

Puzzle 2. Which metric spaces does the Cantor set map onto?

The first one is easy; the second two are well-known… but still, perhaps, not well-known enough!

The answers to Puzzles 1 and 2 can be seen as ‘versal’ properties of the closed unit interval and Cantor set — like universal properties, but without the uniqueness clause.

Posted at November 11, 2017 4:38 AM UTC