This question might appear silly,
I was reading on wikipedia that a closed surface (or manifold in general) is a surface without a boundary, I'd like to elaborate a bit on such definition.
Assuming we have a smooth manifold $\mathcal{S}$ embedded in $\mathbb{R}^n$ can we say that $\mathcal{S}$ is closed if every $p \in \mathcal{S}$ is an interior point in $\mathcal{S}$ w.r.t. the topology induced by $\mathbb{R}^n$ on $\mathcal{S}$?
Unfortunately, there's more to a closed manifold than just being a manifold without boundary.
Your proposal would allow an entire line in the plane. But a line is not compact as a topological space, so it's not a "closed manifold".