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The present invention pertains to an amorphous transition metal oxide that comprises a group-A (Ca, Sr, Ba, and rare-earth elements (RE)) element, cobalt, and a group-B (3d transition element other than Co) element. Observation with a high-resolution transmission electron microscope reveals that this amorphous transition metal oxide has a cluster structure with a particle diameter falling within a range of 0.1-10 nm. The cluster structural part has a γ-CoOOH-type element array structure or an element array structure similar thereto. In the cluster structural part, the molar ratio of the group-A element, cobalt, and the group-B element falls in the range of 0 to 1.99 : 0.01 to 1.0 : 1.0. In a matrix part, the molar ratio of the group-A element, cobalt, and the group-B element falls in the range of not less than 1.0 but less than 2.0 : not less than 1 : 1.0. The present invention also pertains to a water electrolysis positive electrode catalyst and an air electrode catalyst comprising said amorphous transition metal oxide. Disclosed are : an air electrode for a metal-air secondary battery comprising the catalyst mentioned above; and a metal-air secondary battery that has an air electrode comprising the catalyst mentioned above, a negative electrode comprising a negative electrode active material, and an electrolyte interposed between the air electrode and the negative electrode.