Let f : X → X be a map of a compact, connected Riemannian manifold, with or without boundary. For 0$"> ∈ > 0 sufficiently small, we introduce an ∈ -Nielsen number N ∈ ( f ) that is a lower bound for the number of fixed points of all self-maps of X that are ∈ -homotopic to f . We prove that there is always a map g : X → X that is ∈ -homotopic to f such that g has exactly N ∈ ( f ) fixed points. We describe procedures for calculating N ∈ ( f ) for maps of 1 -manifolds.