A closed curve could be the boundary of something, but it has no boundary itself. The boundary of an open curve is its endpoints, a closed curve has no endpoint

A closed curve could be the boundary of something, but it has no boundary itself. The boundary of an open curve is its endpoints, a closed curve has no endpoints. It's important to distinguish having a boundary vs. being the boundary of something, since in some sense the difference between the two is the whole point of homology.