Dot products increase when the two vectors are more similar, decrease (into the negatives) when they are more dissimilar (ie: when the arrow in k space is pointed the opposite direction.).
So this is the set of all vectors x that are less similar to vector p than some measure, B.
In a space visualization, this would draw a decision boundary (set by B) as some hyperplane cutting through k dimensional space. If vector x lies on one side of the boundary, it's in the set A; on the other side, it's not on the set A.
The number of such possible vectors is infinite, this it's an open set.
