I'm reading Hatcher's Algebraic Topology and on page 103 he gives a definition of a $\Delta$-complex which requires, among other things:
A $\Delta$-complex structure on a space $X$ is a collection of maps $\sigma_{\alpha}:\Delta^n \to X$, with $n$ depending on $\alpha$, such that:
(i) The restriction $\sigma|{\mathring{\Delta^n}}$ is injective, and each point $X$ is in the image of exactly one such restriction $\sigma|{\mathring{\Delta^n}}$.
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Among other examples, Hatcher gives this one for a Torus:
I can see how how this has 2 2-simplices ($U$ and $L$), 3 1-simplices ($a$, $b$, and $c$), and 1 0-simplex ($v$). However, I can't see how the point $v$ qualifies under requirement (i) above – it is on the boundary of all six simplices unless I'm misunderstanding something. For $0$-simplices, does the single point qualify as the interior?
Best Answer
On the same page, Hatcher defines the notion of the face of a simplex, and then defines $\mathring{\Delta^n}$ to be $\Delta^n - \partial \Delta^n$, where $\partial \Delta^n$ is the union of the faces of $\Delta^n$. Hatcher's definition of a face implies that $\partial \Delta^0 = \emptyset$. Hence $\mathring{\Delta^0} = \Delta^0$: i.e., the interior of a $0$-simplex is indeed its single point, as you conjectured.
To reconcile this with the topological notion of interior, think of an $n$-simplex as a subspace of its affine span, which is $n$-dimensional. Hatcher's definition then agrees with the topological definition, because when $n=0$, the affine span is a topological space comprising a single point, $x_0$ say, and in that space $\{x_0\}$ is open and is its own interior.