continuity - No bijective continuous $f:(0,1) to [0,1]$ using facts from topology?

In the context of learning topology, I'm triyng to see how to show that there is no bijective continuous function $f:(0,1) \to [0,1]$ using some basic topology facts such as the definition of continuity.

In this setting, $f$ is continuous if for all open $U$ in $[0,1]$, the set $f^{-1}(U)$ is open in $(0,1)$. Equivalently, we could swap "closed" for "open" in this definition.

As a first thought, I don't see any problem with the fact that, if such an $f$ exists, it would satisfy $f^{-1}([0,1]) = (0,1).$ I think this is OK because the set $(0,1)$ is closed in $(0,1)$. Is this correct? If so, what basic topological theorems do we use to prove this statement? Do we need to go to compactness?