Solution. Compactness is a topological property — it is preserved by homeomorphisms, being defined purely in terms of open covers. By Heine–Borel \([0,1]\) is compact, while \((0,1)\) is not: the cover \[U_n = \left (\tfrac 1n,\ 1\right ),\qquad n=2,3,4,\dots \] has no finite subcover, since any finite subfamily has a largest index \(N\) and omits every point below \(1/N\). A homeomorphism would carry a compact space to a compact space, so none exists.

Connectedness cannot settle this question, since both intervals are connected. It is worth noticing which invariant does the work: distinguishing spaces is a matter of finding a topological property one has and the other lacks, and the skill lies in choosing it.