Proof of Dow's reflection theorem without elementary submodels
by
Jerry E. Vaughan
University of North Carolina at Greensboro
Coauthors: R. E. Hodel

A theorem of Alan Dow states that if X is a countably compact space in which every subspace of cardinality at most \aleph₁ is metrizable, then X is metrizable (in Topology Proc Vol. 13, No. 1, 1988). His proof uses elementary submodels. We give a new proof of this theorem without the use of elementary submodels. Our proof is a traditional closure argument similar to the proof of the Hajnal-Juhász reflection theorem, which in the special case relevant here, states: if X is any space in which every subspace of cardinality at most \aleph₁ has a countable base, then X has a countable base. A key concept in our proof is the notion of an < \omega₁, \omega > -structure which distills the closure argument used by Hajnal-Juhász, Dow, and others. Let (X, T) be a topological space. A set of the form { < M_\alpha, L_\alpha > :\alpha < \omega₁} is called an < \omega₁, \omega > -structure for (X, T) provided the following conditions hold for all \alpha < \omega₁: (CNTB) M_\alpha is a countable subset of X, and L_\alpha is a countable subset of T (INCR) M_\alpha subset M_\beta and L_\alpha subset L_\beta for all \alpha < \beta < \omega₁ (CONT) If \alpha is a limit ordinal then M_\alpha = \cup _{\beta < \alpha}M_\beta, and L_\alpha = \cup _{\beta < \alpha}L_\beta (BASE) {U \cap M:U in L} is a base for the subspace topology on M, where M = \cup _{\beta < \omega1}M_\beta, and L = \cup _{\beta < \omega1}L_\beta.

Date received: February 13, 1998