Separation axioms and lattice equivalence (Q717665)

The main result of the paper under review is that the following statements are equivalent for any \(i\in\{1,2,3\frac 1 2 \}\) - \(X\) and \(T_i(X)\) are lattice equivalent, - \(T_0(X)\) satisfies the \(T_i\) separation axiom. Two topological spaces \(X\) and \(Y\) are said to be \textit{lattice equivalent} if there is a bijective map \(\phi\) from the lattice \(\Gamma(X)\) of all closed sets of \(X\) to the lattice \(\Gamma(Y )\) of all closed sets of \(Y\) such that \(\phi\) and \(\phi^{-1}\) are order preserving maps. Let Top be the category of topological spaces with continuous maps as morphisms and \(Top_i\) for \(i\in\{0,1,2,3\frac 1 2\}\) be the full subcategory of \(Top\) whose objects are \(T_i\)-spaces. Recall that \(Top_i\) is a reflective subcategory of \(Top\), i.e. there exists a universal \(T_i\)-space for every topological space \(X\); we denote it by \(T_i(X)\). The author also shows that some special separation axioms (\(T_{(0,2)},T_{(S,D)},T_{(S,1)},T_{(0,3\frac 1 2)}\)) are lattice-invariant properties but others \(( S,T_{(0,1)},T_{(0,S)},T_{(1,2)},T_{(1,S)},T_{(1,3\frac 1 2)}, T_{(0,D)})\) are not. A topological property \(P\) is said to be lattice invariant if a topological space lattice-equivalent to a topological space with property \(P\) also has property \(P\).