Connected sum of compact manifolds is compact

Statement

Suppose $n$ is a natural number and $M_{1},M_{2}$ are Compact connected manifold (?)s of dimension $n$. In other words, each of $M_{1}$ and $M_{2}$ is both a Compact manifold (?) (and in particular, a Compact space (?)) and a Connected manifold (?) (and in particular, a Path-connected space (?)).

Then, the Connected sum (?) $M_{1}\#M_{2}$ is also a compact manifold.

Related facts

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