Extension Operator for Subspaces of Vector Spaces over the Field $\mathbb{F}_2$

In is proved that the free topological vector space $B(X)$ over the field $\mathbb{F}_2=\{0,1\}$ generated by a stratifiable space $X$ is stratifiable, and therefore, for any closed subspace $F\subset B(X)$ (in particular, for $F=X$) and any locally convex space $E$, there exists a linear extension operator $C(F,E)\to C(B(X),E)$ between spaces of continuous maps.