Let the unit arc be,


$$\{x \in \mathbb{R}^2| x\_1^2 + x\_2^2 =1, x\_1 \geq 0, x\_2 \geq 0\}$$


There is something I found curious about the unit arc which is that,


* It has an empty interior viewed as a subset of $\mathbb{R}^2$
* It has an empty relative interior viewed as a subset of its affine
hull, which is again a subset of $\mathbb{R}^2$ and inherits the open
balls.


But intuitively, the arc is not empty (everything in between the arc can be considered as an interior). Is there any way to define interior for these objects? (unit arc, embedded sphere in the orthant, etc)



How to define "interior" for the unit arc?
This seems well described by the notion of interior $\mathrm{Int\ }M$ of a manifold with boundary $M$.


See the Wikipedia subentry [Manifold: Boundary and interior](https://en.wikipedia.org/wiki/Manifold#Boundary_and_interior).


