Let $X$ be a compact metric space and let $f:X\to X$ be a continuous injective map. Which of the following is true?
a) $f(X)$ is dense in $X$
b) $X$ and $f(X)$ are homeomorphic.
c) There exist $x\in X$ such that $f(x)=x$
For a) $X=[0,1]$ defined by $f(x)=\frac{x}{2}$
c) Is Continuous function on a compact set to compact set have a fixed point? b) Continuous function on a compact set is compact. Is the image homemomorphic to $X$?
Answer for b): YES. $X$ and $f(X)$ are compact metric spaces. The inverse is automatically continuous: If $U$ is open in $X$ then $X\setminus U$ is closed, hence compact. So the image $f(X\setminus U)$ is compact (by continuity) hence closed. Its complement is $f(U)$ so $f(U)$ is open. This proves that $f^{-1}$ is continuous.