We prove several reflection theorems in topology using the standard closure method of construction. In particular, we prove the following reflection theorems about weight, point-weight and metrizability:

Theorem [Hajnal-Juhász] 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 \omega₁ and Y is not metrizable.

Theorem [Hodel and Vaughan] If X is compact T₂ 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 \omega₁ and for every Y \in [X]^{\leq \omega₁}, Y is metrizable, then w(X) \leq \omega₁.

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.

Key Words: reflection theorems, metrizable, density, weight, character, separable, first countable, countable base, compact, initially \kappa-compact, countably compact.

MSC: Primary 54E35, 54A25, 54D30.

This article has been published in revised form. R. E. Hodel\ and\ J. E. Vaughan, Reflection theorems for cardinal functions. Special issue in honor of Howard H. Wicke. Topology Appl. {\bf 100} (2000), no.~1, 47--66; MR 2001b:54004