On a question concerning sharp bases

A sharp base B is a base such that whenever ( B i ) i < ω is an injective sequence from B with x ∈ ⋂ i < ω B i , then { ⋂ i < n B i : n < ω } is a base at x. Alleche, Arhangelʹskiĭ and Calbrix asked: if X has a sharp base, must X × [ 0 , 1 ] have a sharp base? Good, Knight and Mohamad claimed to construct an example of a Tychonoff space P with a sharp base such that P × [ 0 , 1 ] does not have a sharp base. However, the space was not regular. We show how to modify the construction to make P Tychonoff.