Is diagonal map of a smooth manifold a proper map

Question

Let $M$ be a manifold.

Is diagonal map $M\rightarrow M\times M$ a proper map?

Answer 1

Let $d:M\rightarrow M\times M$ be the diagonal map.

Let $C$ be a compact subset of $M\times M$.

Let $p_1:M\times M\rightarrow M$ be the first projection and $p_2:M\times M\rightarrow M$ be the second projection.

Let $a\in d^{-1}(C)$ i.e., $(a,a)\in C$. This says $a\in p_1(C)$ as well as $a\in p_2(C)$.

Thus, $d^{-1}(C)\subseteq p_1(C)\cap p_2(C)$.

As $C$ is compact, so is their images $p_1(C),p_2(C)$ and so is their intersection $p_1(C)\cap p_2(C)$.

As $C$ is closed and $d$ is continuous, $d^{-1}(C)$ is closed.

Closed subset of compact set is compact so $d^{-1}(C)$ is compact.

Thus, $d$ is proper map.

Answer 2

Let $C$ be a compact subset of $M\times M$, $d$ the diagonal map and $p$ the projection on the first factor. Let $D=d^{-1}(C)$, it is closed, $p(C)$ is a compact set containing $D$. This implies that $D$ is compact since it is closed and contained in a compact set since the image of a compact set by a continuous map is compact.