The superfluid transition of thin film of 4He adsorbed on packed fine powder is studied theoretically with emphasis on the role of vortices. In this system the film forms a three-dimensional, multiply-connected network. It is shown that the interaction between vortices is proportional to the distance between vortices like a `string' due to the quantum behavior of the flow, and that the vortices are `confined' at low temperatures. The stiffness of a string connecting two vortices is calculated by renormalization. It is found that the transition takes place with the appearance of free vortices at the transition temperature which depends on the particle diameter.