TECHNOLOGY FOCUS - INORGANIC CHEMISTRY - Preferred Dimension: The crystallite diameter is not less than 1.5x102 (preferably not less than 2.6x102) Angstrom . The main peak intensity of the futile type TiO2, of the anatase type TiO2, or Li2TiO3 is not greater than 3 or the basis that the main peak intensity of the lithium titanate is set at 100. The lithium-titanium composite oxide is in the form of particles having an average particle diameter of not greater than 1 microns. Preferred Component: The lithium-titanium composite oxide contains an element X consisting of sodium and/or potassium. The element X is positioned in a lithium (Li) site of the lithium titanate. The positive electrode contains a compound of LiaNibCocMndO2.a=0.1.1;b=0.1-0.5;c=0.9;d=0.1-0.5.Preferred Composition: The amount of the element X contained in the lithium-titanium composite oxide is not less than 0.10-not greater than 3.04 wt.%.    TECHNOLOGY FOCUS - ORGANIC CHEMISTRY - Preferred Component: The non-aqueous electrolyte contains greater than or equal to 3 solvents of propylene carbonate, ethylene carbonate, gamma-butyrolactone, or vinylene carbonate. EXAMPLE - Anatase type titanium oxide was put in a solution prepared by dissolving lithium hydroxide, sodium hydroxide (0.0112 g) and potassium hydroxide (0.1336 g) in pure water, and drying the reaction system. The dried reaction system was baked at 780degreesC for 10 hours to obtain a lithium-titanium composite oxide having a spinel structure, sodium (0.03 wt.%), and potassium (0.42 wt.%). The crystallite diameter of the lithium-titanium composite oxide was 582 Angstrom . A laminate structure was prepared by disposing a positive electrode, a separator formed of a porous polyethylene film (25 microns thick), a negative electrode and another separator one upon the other, followed by spirally winding the laminate structure. The wound laminate structure was pressed under heat of 90degreesC to obtain a flattened electrode group, which was housed in a pack formed of a laminate film, and vacuum dried at 80degreesC for 24 hours. A liquid non-aqueous electrolyte was prepared by dissolving lithium boron tetrafluoride as an electrolyte in a mixed solvent prepared by mixing ethylene carbonate with gamma-butyrolactone. The electrolyte (1.5 mol/L) was dissolved in the mixed solvent. After the liquid non-aqueous electrolyte was poured into a laminate film pack having the housed electrode group, the pack was heat sealed to manufacture a non-aqueous electrolyte secondary battery.