Let \(X\) be a Hausdorff topological space and \(\mathfrak B\) be a family of real-valued functions on \(X\), bounded from below, and this family is endowed with a complete metric. E.g. \(\mathfrak B\) could be the family of all lower semicontinuous functions bounded from below on \(X\), or the family of continuous ones, equipped with the uniform metric. A number of results exist in the literature asserting that almost all (in the Baire category sense) of the functions \(f\) in \(\mathfrak B\) generate minimization problems (i.e. find a minimum of \(f\) in \(X\)) which are well-posed. A function \(f\in{\mathfrak B}\) (or the minimization problem generated by \(f\) in \(X\)) is called well-posed (in the sense of Tykhonov) if \(f\) has a unique minimum in \(X\) and, moreover, every minimizing sequence \((x_ n)\) for \(f\) (this means \(f(x_ n)\to\inf_ X f)\) converges to this unique minimum.
A possible reservation one might have regarding such results is that there may exist distinct functions \(f,g\in{\mathfrak B}\) which have the same values around the common minimum, thus giving, in essence, the same minimization problem. This reservation is mentioned in a previous work of G. Beer [Nonlinear Anal., Theory Methods Appl. 12, No. 6, 647-655 (1988; Zbl 0686.90042)], where, to avoid it, he puts such problems in one and the same equivalence class and proves the above-mentioned generic result in the corresponding quotient space.
In the present paper this idea is broadened by introducing a new equivalence relation in \(\mathfrak B\) which puts in one equivalence class functions with one and the same set of minimizing sequences. The properties of this relation as well as of the corresponding quotient space and quotient mapping are investigated in the paper. Results (generalizing existing ones) saying that almost all (in the Baire category sense) of the equivalence classes generate well-posed minimization problems are proved.