An introduction to reflection theorems on cardinal functions
by
Jerry E. Vaughan
UNC-Greensboro

By a reflection theorem, we mean a theorem which

says that if a space X has a property then some ``small'' subspace

has that property, or by contrapositive, if every ``small'' subspace fails

to have a property, then the whole space X fails to have that property.

The goal of these lectures is to demonstrate the closure method

(which is the heart of many arguments

in the theory of cardinal functions) for

proving some major reflection theorems on

cardinal functions.

The proofs do not use elementary submodels.

We will review the cardinal functions we need, discuss the closure method,

and use the method to prove several theorems.

Here are three of the theorems we prove (where w(X) denotes the weight of

a space X, and

d(X) denotes the density).

Theorem 1 (A. Hajnal and I. Juhász [Proc. Amer. Math. Soc. 79 (1980) 657-658]).

If w(X) >= \kappa then there exists Y subset X with |Y| <= \kappa such that w(Y) >= \kappa.

Theorem 2 (A. Dow [Topology Proceedings 13 no. 1, (1988) 17 -72]).

If X is a countably compact space,

and X is not metrizable, then there exists Y subset X with |Y| <= \omega₁

such that Y is not metrizable.

Theorem 3 (J. Vaughan [Topology Proceedings, to appear]). If d(X) <= \omega₁

and if every

Y subset X with |Y| <= \omega₁ is metrizable, then w(X) <= \omega₁.

We also prove some generalizations of Theorems 2 and 3.

These generalizations are joint work with Richard E. Hodel.

Date received: August 7, 1998