Having a small weight is determined by the small subspaces

Abstract:

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