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We study the notions of weak partial $b$-metric space and weak partialHausdorff $b$-metric space. Moreover, we intend to generalize Nadler's theoremin weak partial $b$-metric space by using weak partial Hausdorff $b$-metricspaces. A non-trivial example to show the validity of our result is given.

Weak partial $b$-metric space and Nadler's theorem