Schubert cells give the stable Grassmannian CW structure

For a symbol $a$, replace the normalized echelon rows by the unique orthonormal echelon frame $(v_1,\dots ,v_n)$ whose last nonzero coordinate is $a_i$ and is nonnegative real. Thus $v_i$ lies in a closed hemisphere $H_i$ of dimension $a_i-1$ over $\mathbb{R}$ or $2a_i-2$ over $\mathbb{C}$. The space $D(a)$ of these mutually orthogonal frames is a closed ball of real dimension $d(a)$ or $2d(a)$: project to $v_1\in H_1$, rotate $v_1$ to $e_{a_1}$ while fixing its orthogonal complement, identify the fiber with the analogous construction for $(a_2-1,\dots ,a_n-1)$, and induct on $n$. For $n=0$ it is a point.

The span map ${\chi }_a:D(a)\to {\mathrm{Gr}}_n({\mathbb{F}}^N)$ restricts on the interior, where every last pivot coordinate is positive, to the pivot-coordinate homeomorphism onto $e(a)$ from [F1]. On the boundary at least one last pivot coordinate is zero; echelon reduction then lowers at least one pivot and never increases another. Hence ${\chi }_a(\mathrm{\partial }D(a))$ lies in the union of cells $e(b)$ with $b_i\le a_i$ for all $i$ and $b\mathrm{\ne }a$, all of smaller dimension.

Induct on the real cell dimension. The zero-dimensional cells form a finite discrete CW complex. If the union $X_r$ of cells of dimension at most $r$ has the asserted CW structure, attach the finitely many disks $D(a)$ of dimension $r+1$ by the boundary maps in step 2.1. The resulting finite CW complex maps continuously and bijectively to the union $X_{r+1}$. Its source is compact, while the Grassmannian is Hausdorff because distinct planes are separated by a squared projection-length function; therefore the map is a homeomorphism. This completes the dimension induction and proves the finite CW structure and closure order.

Under the coordinate inclusion ${\mathbb{F}}^N\subseteq {\mathbb{F}}^{N+1}$, the cells with $a_n\le N$ retain the same symbols and characteristic disks, so they form a subcomplex. The stable Grassmannian has the weak topology with respect to these stages; hence [F2] identifies their union as the asserted CW complex. If a subcomplex has dimension at most $r$, then [F1] gives $a_i-i\le d(a)\le r$, hence $a_i\le i+r$ for every symbol it uses. Only finitely many such symbols exist and all have $a_n\le n+r$, so the subcomplex lies in the finite stage ${\mathrm{Gr}}_n({\mathbb{F}}^{n+r})$.