Surfaces of infinite-type are non-Hopfian

We show that finite-type surfaces are characterized by a topological analog of the Hopf property. Namely, an oriented surface $\mathrm{\Sigma }$ is of finite-type if and only if every proper map $f\text{\hspace{0.17em}⁣}:\mathrm{\Sigma }\to \mathrm{\Sigma }$ of degree one is homotopic to a homeomorphism.