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 $i$ ?

  • $\begingroup$ 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

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 $\Delta^n$.

  • $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.