Having a small weight is determined by the small subspaces

Abstract

We show that for every cardinal κ > ω \kappa > \omega and an arbitrary topological space X if we have w ( Y ) > κ w(Y) > \kappa whenever Y ⊂ X Y \subset X and | Y | ⩽ κ |Y| \leqslant \kappa then w ( X ) > κ w(X) > \kappa as well. M. G. Tkačenko proved this for T 3 {T_3} spaces in [2]. We also prove an analogous statement for the π \pi -weight if κ \kappa is regular.