TITLE: X is a compact metric space, which of the following must be true?
QUESTION [5 upvotes]: $X$ is a compact metric space, $f$ is a continuous function from $X$ $\rightarrow$ $X$, which of the following must be true?
  A. $f$ has fixed point
  B. $f$ is a closed map
  C. $f$  is uniformly continuous

I know C is right by compactness. What about the others?

REPLY [6 votes]: For a non-discrete counterexample to A, consider $X=S^1$ and think about rotations. More generally, any nontrivial rotation of $\mathbb R^n$, with $n$ odd, about the origin has only the origin as a fixed point. So take $X=\mathbb D_1 -D_2$ with D_1 a closed disk about the origin and $D_2$ an open disk of smaller radius about the origin.