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.