Examples of 𝐵(𝐷,𝜆)-refinable and weak \overline𝜃-refinable spaces

In 1980, J. C. Smith asked for examples which would demonstrate the relationships between the properties B ( D , λ ) B(D,\lambda ) -refinability, B ( D , ω 0 ) B(D,{\omega _0}) -refinability, and weak θ ¯ \bar \theta -refinability. This paper gives such examples in the class of T 4 {T_4} spaces.