Extension operator for subspaces of vector spaces over the field \(\mathbb{F}_2\) (Q2080918)

A topological space \(X\) is \textit{stratifiable} if there is a function \(G\) which assigns to each \(n\in \mathbb{N}\) and closed set \(H\subset X\) an open set \(G_n(H)\) containing \(H\) such that \begin{itemize} \item[1.] \(H=\bigcap_n G_n(H)\); \item[2.] if \(H\subset K\), then \(G_n(H)\subset G_n(K)\); \item[3.] \(H=\bigcap_n \overline{G_n(H)}\). \end{itemize} The \textit{free Boolean topological group \(B(X)\)} of a Tychonoff space \(X\) is the vector space generated by \(X\), over the field \(\mathbb{F}_2\), endowed with the coarsest topology with the property that any continuous map \(f: X\to B\), where \(B\) is any Boolean topological group over the field \(\mathbb{F}_2\), extends to a continuous homomorphism \(\hat{f}: B(X)\to B\). The paper under review is devoted to a proof of the following result: For any stratifiable space \(X\), the free Boolean topological group \(B(X)\) is stratifiable.