Finite subset spaces of graphs and punctured surfaces.
We show that the study of topological T0-spaces with a finite number of points agrees essentially with the study of polyhedra, by means of the geometric realization of finite spaces. In this paper all topological spaces are assumed to be T0.
Using the theory of resolving classes, we show that if X is a CW complex of finite type such that for all sufficiently large n, then map⁎(X,K) ∼ ∗ for every simply-connected finite-dimensional CW complex K; and under mild hypotheses on π₁(X), the same conclusion holds for all finite-dimensional complexes K. Since it is comparatively easy to prove the former condition for X = Bℤ/p (we give a proof in an appendix), this result can be applied to give a new, more elementary proof of the Sullivan conjecture....