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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 $i$ ?

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

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

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