On chain conditions in moore spaces

Abstract

A space Y has the countable chain condition (CCC) (discrete countable chain condition (DCCC)) provided each (discrete) collection of mutually exclusive open sets in Y is countable. In [16], the author showed the DCCC, the CCC and separability to be equivalent conditions in completable Moore spaces. However, in [18], Rudin gave an example of a non-separable Moore space with the CCC. And in [15], under the assumption of the Continuum Hypothesis, the author gave an example of a Moore space with the DCCC but not the CCC. In this paper, the author gives an example of a Moore space S with the DCCC but not the CCC which requires no set-theoretic assumptions other than the Axiom of Choice. The construction of this space is of a general nature (to each first-countable T 3 space are associated two Moore spaces), which the author believes will be a useful technique in the search for other counterexamples. The space S is also shown to be a pseudonormal Moore space which does not have the three link property. In addition, the author gives characterizations of chain conditions in Moore spaces and relates these conditions to Moore closure and pseudocompactness.