Show that the subspace \((a, b)\) of \(\mathbb{R}\) is homeomorphic with \((0, 1)\) and the subspace \([a, b]\) of \(\mathbb{R}\) is homeomorphic with \([0, 1]\).
Solution
Assume \(a<b\). The affine map \[ f:[a,b]\longrightarrow[0,1],\qquad f(x)=\frac{x-a}{b-a} \] is continuous, bijective, and has continuous inverse \[ f^{-1}(t)=a+(b-a)t. \] Hence \([a,b]\cong[0,1]\).
Restricting the same formulas gives mutually inverse continuous maps \[ (a,b)\longleftrightarrow(0,1), \] so \((a,b)\cong(0,1)\) as well.