Prove this set is closed Prove that the following set is convex, closed and bounded:
$$\left\{x: x=(x_{1},\dots , x_{n}) \land (\forall i, x_{i}\geq0) \land \sum_{i=1}^{n}x_{i}=1\right\}.$$  
I've been able to prove it is convex and bounded, but I can't seem to use its compliment to show that the compliment is open. Any suggestions would be very helpful. Thank you.

Hint 
Define $f: \mathbb{R}_+^n \rightarrow \mathbb{R}$ by $f:(x_1,...x_n) \mapsto \sum_{i=1}^{n}x_{i}$
Is $f$ continuous ?
Then try to write this set as $f^{-1}(F)$ where $F$ is a closed set.
