proper maps to locally compact spaces are closed in nLab

Proof

Let C⊂XC \subset X be a closed subset. We need to show that every y∈Y\f(C)y \in Y \backslash f(C) has an open neighbourhood U y⊃{y}U_y \supset \{y\} not intersecting f(C)f(C) (by this prop.).

By local compactness of (Y,τ Y)(Y,\tau_Y) (def. ), yy has an open neighbourhood V yV_y whose topological closure Cl(V y)Cl(V_y) is compact. Hence since ff is proper, also f −1(Cl(V y))⊂Xf^{-1}(Cl(V_y)) \subset X is compact. Then also the intersection C∩f −1(Cl(V y))C \cap f^{-1}(Cl(V_y)) is compact, and hence so is

f(C∩f −1(Cl(V y)))=f(C)∩(Cl(V))⊂Y. f(C \cap f^{-1}(Cl(V_y))) = f(C) \cap (Cl(V)) \; \subset Y \,.

This is also a closed subset, since compact subspaces of Hausdorff spaces are closed. Therefore

U y≔V y\(f(C)∩(Cl(V y)))=V y\f(C) U_y \coloneqq V_y \backslash ( f(C) \cap (Cl(V_y)) ) = V_y \backslash f(C)

is an open neighbourhod of yy not intersecting f(C)f(C).