I tried to define an analog of manifold with\without boundary for metric space and I wondered if this definition or a similar one exists in the literature. Let $\left(X,d\right)$ be a connected metric space we say that
it is without "boundary" if there exists an $r\in \mathbb{R}$ s.t. for every $x,y\in X$ there exists a $z\in X$ s.t. $d\left(z,x\right)=r$ and $d\left(z,y\right)=d\left(z,x\right)+d\left(x,y\right)$.
Under this definition a sphere with the intrinsic metric is a space without boundary while a disc is space with boundary. Note: We may want want to include the condition that the metric space is a length metric space in the definition.
Define a metric $d$ on $\mathbb{S}^2$ : $d(x,y)=\sqrt{d_0(x,y)}$ where $d_0$ is canonical metric on $\mathbb{S}^2$. It is not intrinsic metric.
Consider $X$, which is Euclidean cone over $ \mathbb{S}^2$ as a set, s.t. $$d_X((s,x),(t,y))=|s-t|+d(x,y) $$