Sharp bases and weakly uniform bases versus point-countable bases

✍ Scribed by A.V. Arhangel'skiı̌; W. Just; E.A. Rezniczenko; P.J. Szeptycki

• Publisher: Elsevier Science
• Year: 2000
• Tongue: English
• Weight: 86 KB
• Volume: 100
• Category: Article
• ISSN: 0166-8641
• DOI: 10.1016/s0166-8641(98)00136-9

✦ Synopsis

A base B for a topological space X is said to be sharp if for every x ∈ X and every sequence (U n ) n∈ω of pairwise distinct elements of B with x ∈ U n for all n the set { i<n U i : n ∈ ω} forms a base at x. Sharp bases of T 0 -spaces are weakly uniform. We investigate which spaces with sharp bases or weakly uniform bases have point-countable bases or are metrizable. In particular, Davis, Reed, and Wage had constructed in a 1976 paper a consistent example of a Moore space with weakly uniform base, but without a point-countable base. They asked whether such an example can be constructed in ZFC. We partly answer this question by showing that under CH, every first-countable space with a weakly uniform base and at most ℵ ω isolated points has a point-countable base.