Definition
A space X is locally weakly pseudocompact at a point x in X if there is a basic system of open neighborhoods N of x whose all of elements are weakly pseudocompact. If X is locally weakly pseudocompact at every point, X is simply called locally weakly pseudocompact.
Reference
O. Okunev, A. Tamariz-Mascarua, Generalized linearly ordered spaces and weak pseudocompactness, Comment.Math.Univ.Carolin. 38,4 (1997)775–790.