Manifold boundaries

• How many boundary components may a manifold with metrisable interior have?
• There may be only countably many compact boundary components.
• There may be continuum many non-Lindelöf boundary components.
• Continuum many boundary components may be replaced by trees.
• The trees must have weight at most ω1ω1 and their vertices must have finite order.

Given a manifold M, we consider properties of the frontier of a dense metrisable submanifold N of M. We show that if each boundary component of a manifold-with-boundary is compact and the interior is metrisable, then there are only countably many boundary components and the manifold itself is metrisable. We also show how to construct manifolds by adding to a metrisable manifold continuum many sets, such as graphs whose vertices are trees.