  We study the phase structure of a surface model by using the canonical Monte
Carlo simulation technique on triangulated, fixed connectivity, and spherical
surfaces with many fine holes. The size of a hole is assumed to be of the order
of lattice spacing (or bond length) and hence can be negligible compared to the
surface size in the thermodynamic limit. We observe in the numerical data that
the model undergoes a first-order collapsing transition between the smooth
phase and the collapsed phase. Moreover the Hasudorff dimension H remains in
the physical bound, i.e., H<3 not only in the smooth phase but also in the
collapsed phase at the transition point. The second observation is that the
collapsing transition is accompanied by a continuous transition of surface
fluctuations. This second result distinguishes the model in this paper and the
previous one with many holes, whose size is of the order of the surface size,
because the previous surface model with large-sized holes has only the
collapsing transition and no transition of surface fluctuations.
