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!


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