We study the evolution of the superconducting state in a perforated disk by varying the size of the hole. The superconducting properties are investigated by means of transport measurements around the superconducting/normal phase boundary T-c(H). A transition from a one-dimensional to a two-dimensional regime is seen when increasing the magnetic field for disks with small holes. A good agreement is found between the measured T-c(H) line and the calculations performed in the framework of the linearized Ginzburg-Landau theory. The effect of breaking the axial symmetry of the structure by moving the hole away from the center of the disk is also studied. An enhanced critical field is found for the asymmetric structures, possibly due to the recovery of the singly connected state.