NOVELTY - A fine fiber structure comprises a porous fine fiber layer containing a polymeric fine fiber having average diameter of 50-3000 nm. The fine fiber structure has penetration rate of more than 20 cm2/minute. The porous fine fiber layer has mean pore diameter of 0.01-15 mu m, thickness of 0.0025-0.3 mm, porosity of 20-90%, basic weight of 1-90 g/m2, fragile permeability of less than 46 m3/minute/m2, and numbers of Macmillans of 2-15.    USE - Fine fiber structure is used for battery e.g. lithium battery, lithium-ion battery, alkali battery and lithium ion gel polymer battery, electrical double layer capacitor and condenser (all claimed).    ADVANTAGE - The fine fiber structure has excellent ionic permeability, short circuit-proof property and productivity.    DETAILED DESCRIPTION - A fine fiber structure comprises a porous fine fiber layer containing a polymeric fine fiber having average diameter of 50-3000 nm. The fine fiber structure has penetration rate of more than 20 cm2/minute, which is calculated by cutting fiber structure to dimension of 6 cmx 8 cm, pinching obtained structure on the upper and lower sides of glass plate having 2 sheets of 20 cmx 30 cm and thickness of 5 mm, mounting, such that contact-pressure is affected to a dimension of 0.1 kgf/cm2 on the upper glass plate. The porous fine fiber layer has mean pore diameter of 0.01-15 mu m, thickness of 0.0025-0.3 mm, porosity of 20-90%, basic weight of 1-90 g/m2, fragile permeability of less than 46 m3/minute/m2, and numbers of Macmillans of 2-15. The dimension of one side of up-and-down glass plate is 6 cmx 1 cm. The overflowing portion is immersed in an electrolyte solution bath. The area (cm2) in which electrolyte solution is permeated, of test sample interposed into the glass plate for 1 minute is measured. Electrolyte solution is measured at 25 degrees C using the liquid mixture of ethylene carbonate/ethylmethyl carbonate. The weight ratio of ethylene carbonate and ethylmethyl carbonate is 3/7. INDEPENDENT CLAIMS are included for the following:    (1) battery;    (2) electrical double layer capacitor; and    (3) condenser.