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.