A polymer derived from a polyimide copolymer, wherein the polymer derived from the polyimide copolymer includes picopores, said picopores having an average diameter of 100 pm to 1000 pm, measured by positron annihilation lifetime spectroscopy (PALS), and the polyimide copolymer comprises a repeating unit represented by any one of the following Chemical Formulae 37 to 40: wherein, in the above Chemical Formulae 37 to 40, Ar 1 is an aromatic group selected from a substituted or unsubstituted quadrivalent C6 to C24 arylene group and a substituted or unsubstituted quadrivalent C4 to C24 heterocyclic group, where the aromatic group is present singularly; at least two aromatic groups are fused to form a condensed cycle; or at least two aromatic groups are linked by a single bond or a functional group selected from O, S, C(=O), CH, S(=O) 2, Si(CH 3) 2, (CH 2) p, where 1≤p≤10, (CF 2) q, where 1≤q≤10, C(CH 3) 2, C(CF 3) 2, or C(=O)NH, Ar 2 is an aromatic group selected from a substituted or unsubstituted divalent C6 to C24 arylene group and a substituted or unsubstituted divalent C4 to C24 heterocyclic group, where the aromatic group is present singularly; at least two aromatic groups are fused to form a condensed cycle; or at least two aromatic groups are linked by a single bond or a functional group selected from O, S, C(=O), CH, S(=O) 2, Si(CH 3) 2, (CH 2) p, where 1≤p≤10, (CF 2) q, where 1≤q≤10, C(CH 3) 2, C(CF 3) 2, or C(=O)NH, wherein the term "substituted" means substituted with at least one substituent selected from the group consisting of a C1 to C10 alkyl group, a C1 to C10 alkoxy group, a C1 to C10 haloalkyl group, and a C1 to C10 haloalkoxy group, Q is O, S, C(=O), CH, S(=O) 2, Si(CH 3) 2, (CH 2) p, where 1≤p≤10, (CF 2) q, where 1≤q≤10, C(CH 3) 2, C(CF 3) 2, C(=O)NH, C(CH 3)(CF 3), or a substituted or unsubstituted phenylene group, where the substituted phenylene group is a phenylene group substituted with a C1 to C6 alkyl group or a C1 to C6 haloalkyl group, where the Q is linked with aromatic groups with m-m, m-p, p-m, or p-p positions, Y is the same or different in each repeating unit and is independently selected from OH, SH, or NH 2, m is an integer ranging from 10 to 400, l is an integer ranging from 10 to 400, wherein the polymer is doped with an acid dopant, and wherein the acid dopant includes one selected from the group consisting of sulfuric acid, hydrochloric acid, phosphoric acid, nitric acid, HBrO 3, HClO 4, HPF 6, HBF 6, 1-methyl-3-methylimidazolium cations (BMIM +), and a combination thereof. Ar 1 is an aromatic group selected from a substituted or unsubstituted quadrivalent C6 to C24 arylene group and a substituted or unsubstituted quadrivalent C4 to C24 heterocyclic group, where the aromatic group is present singularly; at least two aromatic groups are fused to form a condensed cycle; or at least two aromatic groups are linked by a single bond or a functional group selected from O, S, C(=O), CH, S(=O) 2, Si(CH 3) 2, (CH 2) p, where 1≤p≤10, (CF 2) q, where 1≤q≤10, C(CH 3) 2, C(CF 3) 2, or C(=O)NH, Ar 2 is an aromatic group selected from a substituted or unsubstituted divalent C6 to C24 arylene group and a substituted or unsubstituted divalent C4 to C24 heterocyclic group, where the aromatic group is present singularly; at least two aromatic groups are fused to form a condensed cycle; or at least two aromatic groups are linked by a single bond or a functional group selected from O, S, C(=O), CH, S(=O) 2, Si(CH 3) 2, (CH 2) p, where 1≤p≤10, (CF 2) q, where 1≤q≤10, C(CH 3) 2, C(CF 3) 2, or C(=O)NH, wherein the term "substituted" means substituted with at least one substituent selected from the group consisting of a C1 to C10 alkyl group, a C1 to C10 alkoxy group, a C1 to C10 haloalkyl group, and a C1 to C10 haloalkoxy group, Q is O, S, C(=O), CH, S(=O) 2, Si(CH 3) 2, (CH 2) p, where 1≤p≤10, (CF 2) q, where 1≤q≤10, C(CH 3) 2, C(CF 3) 2, C(=O)NH, C(CH 3)(CF 3), or a substituted or unsubstituted phenylene group, where the substituted phenylene group is a phenylene group substituted with a C1 to C6 alkyl group or a C1 to C6 haloalkyl group, where the Q is linked with aromatic groups with m-m, m-p, p-m, or p-p positions, Y is the same or different in each repeating unit and is independently selected from OH, SH, or NH 2, m is an integer ranging from 10 to 400, l is an integer ranging from 10 to 400, wherein the polymer is doped with an acid dopant, and wherein the acid dopant includes one selected from the group consisting of sulfuric acid, hydrochloric acid, phosphoric acid, nitric acid, HBrO 3, HClO 4, HPF 6, HBF 6, 1-methyl-3-methylimidazolium cations (BMIM +), and a combination thereof.