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Abstract. Natural weakenings of uniformizability of a ladder system on ω 1 are considered. It is shown that even assuming CH all the properties may be distinct in a strong sense. In addition, these properties are studied in conjunction with other properties inconsistent with full uniformizability, which we call anti-uniformization properties. The most important conjunction considered is the uniformization property we call countable metacompactness and the anti-uniformization property we call thinness. The existence of a thin, countably metacompact ladder system is used to construct interesting topological spaces: a countably paracompact and nonnormal subspace of ω 2 1 , and a countably paracompact, locally compact screenable space which is not paracompact. Whether the existence of a thin, countably metacompact ladder system is consistent is left open. Finally, the relation between the properties introduced and other well known properties of ladder systems, such as ♣, is considered.

Abstract. Hušek defines a space X to have a small diagonal if each uncountable subset of X 2 disjoint from the diagonal has an uncountable subset whose closure is disjoint from the diagonal. Hušek proved that a compact space of weight ω 1 which has a small diagonal will be metrizable, but it remains an open problem to determine if the weight restriction is necessary. It has been shown to be consistent that each compact space with a small diagonal is metrizable; in particular, Juhász and Szentmiklóssy proved that this holds in models of CH. In the present paper we prove that this also follows from the Proper Forcing Axiom (PFA). We furthermore present two (consistent) examples of countably compact non-metrizable spaces with small diagonal, one of which maps perfectly onto ω 1 .1. Introduction. We refer the reader to Gruenhage's interesting article [Gru02] for more background on spaces with small diagonal. In particular, as mentioned there, H. X. Zhou [Zho82] is responsible for broadening the question to countably compact and Lindelöf spaces, while Hušek originally asked about compact and ω 1 -compact spaces. It has already been shown to hold in some models that compact spaces with small diagonal are metrizable (see [Zho82,Dow88a,Dow89,JS92]). As mentioned in the abstract, we prove that PFA implies that compact spaces with small diagonal are metrizable. This is the content of the second section. In the third section, we present two constructions of countably compact spaces with small diagonals. The first, from the hypothesis ♦ + , maps perfectly onto ω 1 with metric fibers. The second example is presented because the set-theoretic hypothesis that we are able to use is quite weak.In this section we review some of the already established results concerning spaces with small diagonal that will be useful in our proofs. It is easily seen that a space with a G δ -diagonal has a small diagonal. The Sorgenfrey line is a well known example of a Lindelöf space with a G δ -diagonal which

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The finite derived set property asserts that any infinite subset of a space has an infinite subset with only finitely many accumulation points. Among other things, we study this property in the case of a function space with the topology of pointwise convergence.2000 AMS Classification: 54A25, 54A35, 54D55.