Connected sum of compact manifolds is compact

Statement
Suppose $$n$$ is a natural number and $$M_1, M_2$$ are fact about::compact connected manifolds of dimension $$n$$. In other words, each of $$M_1$$ and $$M_2$$ is both a fact about::compact manifold (and in particular, a fact about::compact space) and a fact about::connected manifold (and in particular, a fact about::path-connected space).

Then, the fact about::connected sum $$M_1 \# M_2$$ is also a compact manifold.

Similar facts

 * Connected sum of simply connected manifolds is simply connected
 * Fundamental group of connected sum is free product of fundamental groups in dimension at least three