Fixed Point Property of Rationals

Question

Does the space $X = [0,1] \cap \mathbb{Q}$ have a fixed point property (FPP)? i.e. can we find a continuous function, $f: [0,1] \cap \mathbb{Q} \rightarrow [0,1] \cap \mathbb{Q}$ without a fixed point?

(My version of the answer: Since this set of rationals can be considered homeomorphic to a countable metric space without isolated points (is this correct?), and since this space has no FPP, neither does $[0,1] \cap \mathbb{Q}$)

Answer 1

Hint 1: Go back and make sure you fully understand continuous maps, discrete spaces, and the topology of $\Bbb Q$ before you address this problem specifically. Go all the way back to the basic definitions and show that:

1. $Q$ is not discrete.
2. $Q\cap[0,1]$ is not discrete
3. $\Bbb Z$ is discrete
4. If $X$ is discrete, every map $f:X\to X$ is continuous
5. A metric space without isolated points is not discrete
6. A finite discrete space with more than one point has a continuous self-map with no fixed points.
7. The same, but for the infinite discrete space $\Bbb Z$.

Hint 2: Think about the problem geometrically. Imagine that you drew a graph of a no-fixed-point function. What would it look like?