Schubert cells give the stable Grassmannian CW structure

For a symbol $a$, each plane of $e(a)$ has a unique orthonormal echelon frame $(v_1,\dots ,v_n)$ with $v_i\in {\mathbb{F}}^{a_i}$ and positive real $a_i$th coordinate: orthonormalize its flag of subspaces of dimensions $1,\dots ,n$. Let $H_i$ instead be the closed hemisphere of unit vectors in ${\mathbb{F}}^{a_i}$ whose $a_i$th coordinate is real and nonnegative, and let $D(a)$ consist of all mutually orthogonal frames with $v_i\in H_i$, allowing zero pivot coordinates. Project to $v_1\in H_1$ and put $v_0=e_{a_1}$. There is a continuous family of orthogonal (real case) or unitary (complex case) maps ${\rho }_v$ carrying $v$ to $v_0$, equal to the identity outside their span and at $v=v_0$: use the plane rotation in the real case and the determinant-one unitary map on their complex two-plane in the complex case. The nonnegative real coordinate ensures $v\mathrm{\ne }-v_0$, and these maps tend to the identity as $v\to v_0$. In particular they fix the common perpendicular of $v$ and $v_0$, rather than $v^{\perp }$ alone. They act only in ${\mathbb{F}}^{a_1}$, so preserve $H_i$ and its boundary for $i>1$. Applying ${\rho }_v$ to the remaining vectors trivializes the projection, with fiber $D(a_2-1,\dots ,a_n-1)$ after deleting coordinate $a_1$. Induction, starting with $n=0$, identifies $D(a)$ with a product of closed balls, hence a closed ball of real dimension $d(a)$ or $2d(a)$. Its boundary is exactly where at least one pivot coordinate is zero.

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 ${\mathrm{Gr}}_n({\mathbb{F}}^{n+r})$.