I'm looking for a way to convergence on subspaces.
If $G_k(\mathbb{R}^m)=\{ W: W$ is subspace of $\mathbb{R}^m, \dim W=k \}$ and consider in $G_k(\mathbb{R}^m)$ one topology $\tau$.
I would like to find a non-trivial topology for the following statement is true:
$\lbrace S_k\rbrace \subset G_n(\mathbb{R}^m)$ converges to $S\in G_n(\mathbb{R}^m)$ sss for every $k\in \mathbb{N}$ there is a basis $\lbrace u^k_1,\ldots,u^k_n\rbrace$ of $S_k$ such that $\lbrace \displaystyle{\lim_{ k \rightarrow +\infty}}u^k_1,\ldots,\displaystyle{\lim_{ k \rightarrow +\infty}}u^k_n\rbrace$ is a basis of $S$.
All suggestions are welcome!
Note: I thought that the appropriate topology was $\tau$ where $U\in \tau$ is open iff the set $\widehat{U}=\lbrace v: v\in W\backslash \lbrace 0\rbrace, \mbox{for some} \ W\in U \rbrace$ is open in $\mathbb{R}^m$ but as shown in the following post is not true.
The topology induced by the metric $$d(X, Y) = \lVert P_X - P_Y \rVert$$ that you mentioned, where $P_X$ and $P_Y$ are respectively the orthogonal projections of $\mathbb R^m$ onto $X$ and $Y$, works.
(Note that this topology is in fact the standard topology mentioned by Peter Franek and others in the comments to the question. If you're curious about the equivalence of the various definition of this topology, you might start with the section in the Wiki article and this question over at MO. But this is beside the point. All we need is that the map $X \mapsto P_X$ is an injection $G_n(\mathbb R^m) \hookrightarrow \mathbb R^{m\times m}$, so that $G_n(\mathbb R^m)$ can inherit the metric topology on $\mathbb R^{m \times m}$ as an embedded subset.)
To see why, we use this nice formula for the projection: If the columns of $A \in \mathbb R^{m \times n}$ form a basis of $X \in G_n(\mathbb R^m)$, then $$P_{X} = A(A^\top A)^{-1} A^\top.$$ Note that this is a continuous function of $A$. Hence, the convergence we have at the level of bases implies the convergence of the corresponding sequence of projections $P_{S_k}$ to $P_S$, and equivalently that of the sequence of spans $S_k$ to $S$.