Polymer network formation via Michael addition of two tetra-arm poly(ethylene glycol) (TetraPEG) prepolymers possessing maleimide (MA) and thiol (SH) terminals was investigated in a typical aprotic ionic liquid containing a basic catalyst. The gelation reaction proceeded successfully at room temperature to give high network connectivity (similar to 98%). The resulting TetraPEG gel showed ideal liquid-like ion conducting properties and was applied to a polymer electrolyte for electric double-layer capacitors (EDLC).