Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Cosmicity of cometrizable spaces


Author: Gary Gruenhage
Journal: Trans. Amer. Math. Soc. 313 (1989), 301-315
MSC: Primary 54E20; Secondary 54A35
DOI: https://doi.org/10.1090/S0002-9947-1989-0992600-5
MathSciNet review: 992600
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A space $ X$ is cometrizable if $ X$ has a coarser metric topology such that each point of $ X$ has a neighborhood base of metric closed sets. Most examples in the literature of spaces obtained by modifying the topology of the plane or some other metric space are cometrizable. Assuming the Proper Forcing Axiom (PFA) we show that the following statements are equivalent for a cometrizable space $ X$ : (a) $ X$ is the continuous image of a separable metric space; (b) $ {X^\omega }$ is hereditarily separable and hereditarily Lindelöf, (c) $ {X^2}$ has no uncountable discrete subspaces; (d) $ X$ is a Lindelöf semimetric space; (e) $ X$ has the pointed $ {\text{ccc}}$. This result is a corollary to our main result which states that, under PFA, if $ X$ is a cometrizable space with no uncountable discrete subspaces, then either $ X$ is the continuous image of a separable metric space or $ X$ contains a copy of an uncountable subspace of the Sorgenfrey line.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 54E20, 54A35

Retrieve articles in all journals with MSC: 54E20, 54A35


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1989-0992600-5
Article copyright: © Copyright 1989 American Mathematical Society