# A property of the n-simplex

Let $$\Delta ^{n}$$ a n-simplex and $$\Delta_{0}^{n-1}, \Delta_{1}^{n-1}, \cdots , \Delta_{n}^{n-1}$$ be the $$(n-1)$$ - faces of $$\Delta^{n}$$.

The subsets $$\Delta^{n} \backslash \Delta_{i}^{n-1}$$ are open for all $$i$$ ?

• No. If you remove an edge from a two-simplex (triangle), the resulting figure is neither open nor closed. – saulspatz Dec 2 '18 at 4:59

## 1 Answer

Yes, these sets are open (in $$\Delta^n$$!) as all (sub)simplices are compact and thus absolutely closed. And $$\Delta^n \setminus \Delta_i^{n-1} = \Delta^n \cap (\mathbb{R}^{n+1}\setminus \Delta_{i}^{n-1})$$, which is the intersection of an open set in the ambient space with $$\Delta^n$$.

• Thanks !! i missed put in $\Delta^{n}$ . – Juan Daniel Valdivia Fuentes Dec 2 '18 at 13:56
• @JuanDanielValdiviaFuentes They are neither open nor closed in the ambient space, but inside the lareg simplex they are open. – Henno Brandsma Dec 2 '18 at 14:00