Let $G_k(\mathbb{R}^m)=\{ W: W$ is subspace of $\mathbb{R}^m, \dim W=k \}$ measurable and suppose that the application $$ \displaystyle{\begin{array}{rccl} h:&Z&\longrightarrow& (G_k(\mathbb{R}^m), \tau)\\ &x&\longmapsto& H_x \end{array} } $$ is continuous(*), where $\tau$ is topology induced by the metric $d(X, Y) = \lVert P_X - P_Y \rVert$, where $P_X$ and $P_Y$ are respectively the orthogonal projections of $\mathbb R^m$ onto $X$ and $Y$, and $Z\subset \mathbb{R}^m$ is open. Then
1) for all $x\in Z$ there is a measurable set $U_x\subset Z$ with $x\in U_x$ and $v_i: U_x \rightarrow V$ measurable for all $1\leq i\leq d(x)$ and such that
2)$\lbrace v_1(z),\ldots,v_{d(x)}(z)\rbrace$ is basis of $H_z$ for all $z\in U_x$.
I would greatly appreciate any suggestions to test this assertion
Note: The statement is also true if if we replace (*) by Borel-measurable and $Z$ Borel - measurable. But I have no idea how to prove