The filling of slits with identical planar walls is investigated in the frame of the density functional theory. For this kind of slit, the confining potential is symmetric with respect to its central plane. Closed and open systems are studied by applying, respectively, the canonical and grand canonical ensembles (CE and GCE). Results obtained for the confinement of fluid Ne by alkaline surfaces are reported. The behavior of these systems is analyzed by varying the strength of the Ne-substrate attraction, the temperature T, and the coverage Γ_l. It is assumed that the one-body density of the fluid, ρ(r), is uniform along the (x, y) planes parallel to the walls, becoming a function of the coordinate z perpendicular to those planes. Two sorts of solutions are found for the density profile: (i) symmetric ones that follow the left-right symmetry of the potential exerted by the walls and (ii) asymmetric ones that break the symmetry of the slit. The pores are wide enough for determining prewetting (PW) lines and wetting and critical PW temperatures, i.e., T_w and T_cpw, from the analysis of symmetric solutions provided by both the CE and GCE schemes. Asymmetric species are examined in detail for T > T_w. It is shown that for a given Ne-substrate pair at a fixed T both the CE and GCE frames yield only one asymmetric 2-fold degenerate stable profile (formed by a “thin” wetting film at one wall and a “thick” wetting film at the other) coexisting with two symmetric profiles (formed by “thin” or “thick” wetting films at the two walls), while the remaining asymmetric states are at best metastable. This feature occurs along PW lines and disappears at T_cpw.