# In- and out-boundaries of the unit interval

I am trying to understand orientation of one-point manifolds in order to figure out in- and out-boundaries of the unit interval. So far I have understood that an orientation of a zero-manifold is an assignment of either $$+1$$ or $$-1$$, since the tangent space of a point is the trivial vector space $$V=\{0\}$$. The unique basis for this vector space is the empty set.

Now let $$\{0\}$$ and $$\{1\}$$ of the unit interval $$I=[0,1]$$ have orientation $$+1$$. Let $$I$$ have orientation with positive basis $$[e_1]$$ induced from the standard orientation on $$\mathbb{R}$$. I've been told, that $$\{0\}$$ then becomes the in-boundary and $$\{1\}$$ the out-boundary, but I don't understand how, this is what I've tried:

In order to find out whether $$\{0\}$$ is an in- or out-boundary of $$I$$, I have to find out whether the positive normals points in to or out from $$I$$. This is where I get stuck. How do I do this, if the basis of the tangent space of $$\{0\}$$ is the empty set?

This is what I am thinking: In order to find out whether a vector $$w$$ on the tangent space $$T_{\{0\}}I$$ is a positive normal, then $$[\emptyset, w]$$ must be a positive basis for $$T_{\{0\}}I$$. But the the orientation $$+1$$ of $$\{0\}$$ doesn't matter, hence if I assign $$-1$$ to both $$\{0\}$$ and $$\{1\}$$ the in- and out-boundaries will not change.

I apologize if this is trivial or silly. Thanks in advance for your help!