Subspace restrictions of homeomorphisms and $[0,1)\not\cong(0,1)$

Suppose \((X,\mathcal T)\) and \((Y,\mathcal T')\) are topological spaces, \(A\subseteq X\), and \[ f:(X,\mathcal T)\longrightarrow(Y,\mathcal T') \] is a homeomorphism. Show that the restriction \[ f|_A:(A,\mathcal T_A)\longrightarrow(f(A),\mathcal T'_{f(A)}) \] is a homeomorphism for the corresponding subspace topologies.

Show that there is no homeomorphism \[ [0,1)\longrightarrow(0,1), \] where both spaces carry the subspace topology inherited from the usual topology on \(\mathbb R\).