The only connected one-dimensional example is a circle.
The sphere, torus, and the Klein bottle are all closed two-dimensional manifolds.
The real projective space RPn is a closed n-dimensional manifold.
The complex projective space CPn is a closed 2n-dimensional manifold.
[1] A line is not closed because it is not compact.
A closed disk is a compact two-dimensional manifold, but it is not closed because it has a boundary.
Every closed manifold is a Euclidean neighborhood retract and thus has finitely generated homology groups.
is a closed connected n-manifold, the n-th homology group
[3] Moreover, the torsion subgroup of the (n-1)-th homology group
This follows from an application of the universal coefficient theorem.
is an isomorphism for all k. This is the Poincaré duality.
For a connected manifold, "open" is equivalent to "without boundary and non-compact", but for a disconnected manifold, open is stronger.
For instance, the disjoint union of a circle and a line is non-compact since a line is non-compact, but this is not an open manifold since the circle (one of its components) is compact.
Most books generally define a manifold as a space that is, locally, homeomorphic to Euclidean space (along with some other technical conditions), thus by this definition a manifold does not include its boundary when it is embedded in a larger space.
However, this definition doesn’t cover some basic objects such as a closed disk, so authors sometimes define a manifold with boundary and abusively say manifold without reference to the boundary.
But normally, a compact manifold (compact with respect to its underlying topology) can synonymously be used for closed manifold if the usual definition for manifold is used.
A line is a closed subset of the plane, and it is a manifold, but not a closed manifold.
The notion of a "closed universe" can refer to the universe being a closed manifold but more likely refers to the universe being a manifold of constant positive Ricci curvature.