Structure theorems for finite unions of subspaces of special kind (Q330068)

After its proof the following result is used to draw various conclusions: for an infinite cardinal \(\tau\) and a \(T_3\) space \(X\) that is the union of a family \(\gamma\) of \(m\) subspaces each of weight at most \(\tau\) there is a collection \(\{F_i\;\mid i=1,\dots,n\}\) of closed subsets of \(X\) with \(F_i\subset F_{i+1}\), \(F_1\) and each \(F_{i+1}\setminus F_i\) has weight at most \(\tau\) and \(n\leq m\). Some sample conclusions: if each member of \(\gamma\) is second countable [locally compact] then every nonempty open subset of \(X\) has a nonempty second countable [locally compact] open subspace; if \(X\) is countably compact and each member of \(\gamma\) has a point-countable base then \(X\) must be compact and finitely sequential and each of \(F_1\) and the sets \(F_{i+1}\setminus F_i\) may be chosen to be metrisable and locally compact.