Theorem [Hajnal-Juh\'asz] If $w(X)\geq\kappa$ then there exists $Y\subset X$ such that $|Y|\leq\kappa$ and $w(Y)\geq\kappa$.

Theorem [Dow] If $X$ is countably compact and not metrizable, then there exists $Y\subset X$ such that $|Y|\leq\omega_1$ and $Y$ is not metrizable.

Theorem [Hodel and Vaughan] If $X$ is compact $T_2$ and $pw(X)\geq\kappa$ then there exists $Y\subset X$ such that $|Y|\leq\kappa$ and $pw(Y)\geq\kappa$.

Theorem [Vaughan] If $d(X)\leq\omega_1$ and for every $Y\in[X]^{\leq\omega_1}$, $Y$ is metrizable, then $w(X)\leq\omega_1$.

This paper is based on joint work with Richard E. Hodel, and is a slightly revised version of three lectures given to The Second Galway Topology Colloquium, September 2 - 5, 1998 at The University of Oxford, Oxford, United Kingdom. It is intended to be accessible to graduate students.