First I'll state my current mathematical knowledge on the subject of homeomorphisme: very low. I'm doing a class of dynamical systems and I got the following question:
Let $f:[0,1] \rightarrow (2,8)$ and $f$ is one-to-one and continuous. Now show $f$ is not onto.
So I started of by stating that $f$ is either continuously increasing or continuously decreasing, that's because $f$ is one-to-one. Now intuitively it makes sense that $f$ cannot be onto since it maps a closed interval into a open one, but I don't know how to show/prove this.
The exercise has a hint that $[0,1]$ and $(2,8)$ are not homeomorphic but I have no idea how to use this.