This paper explores two fundamental concepts: branch width and weak ultrafilter. Branch width is a significant graph width parameter that measures the degree of connectivity in a graph using a branch decomposition and a symmetric submodular function. Weak ultrafilter, introduced as a weakened definition of an ultrafilter, plays a vital role in interpreting defaults in logic. We introduce the concept of Weak Ultrafilter on the connectivity system (X, f) and demonstrate its duality with branch decomposition. This study enhances our understanding of these concepts in graph combinatorial and logical contexts.