Metrization and stratification of squares of topological spaces (Q1379791)

A topological space \(X\) is said to have the property of Weak Assigned Separation (WAS) if for every countable family \({\mathcal U} =\{U_i \mid i<\omega\}\) of pairwise disjoint open subsets of \(X\) there exists, for each \(i<\omega\), a countable family of mappings \((V_{i,n}: U_i\to \wp(U_i))_{n< \omega}\) such that: (i) each \(V_{i,n}\) is a neighborhood assignment on \(U_i\), i.e. \(V_{i,n}(x)\) is a neighbourhood of \(x\) for each \(x\in U_i\); (ii) each \(V_{i,n}\) is topologically symmetric, i.e. for each \(x\in U_i\) the set \(\{y\in U_i\mid x\in V_{i,n} (y)\}\) is a neighborhood of \(x\); (iii) for each \(p\notin \bigcup\{\text{cl} U_i\mid i< \omega\}\) and for each neighborhood \(W\) of \(p\) there exists a neighborhood \(U\) of \(p\), an infinite subset \(I\) of \(\omega\), and, for each \(i\in I\), an \(n(i) <\omega\) such that \(U\cap V_{i,n(i)} (x)= \emptyset\) whenever \(i\in I\) and \(x\in U_i\smallsetminus W\). It is shown that a Hausdorff space \(X\) containing a copy of \(\omega+1\) is metrizable if and only if the square \(X^2\) has the property of Weak Assigned Separation. If one drops condition (ii) in the definition of property WAS one obtains a similar characterization of stratifiable spaces.