Having a hard proving $\{ t_i > 0 \mid \sum t_i = 1\}$ is open

Question

Let $S = \{ (t_0, ..., t_n) \in \mathbb{R}^{n+1} \mid \forall i, t_i >0, \sum t_i = 1\}$. Is the set $S$ closed or open or neither ?

I think the set $S$ is closed, yet it's hard for me to prove it.Here is my try : Let $f: \mathbb{R}^{n+1} \to \mathbb{R}^{n+2} : (t_0,...,t_n) \mapsto (t_0,t_1,...,t_n, \sum t_i)$. Then since we know that the $t_i$ can't be equal to $1$ since it would imply that the other $t_i$ are $0$, we have that :

$$S = f^{-1}(]0,1[, ..., ]0,1[, \{1\})$$

Yet here I I think I can't conclude since $(]0,1[, ..., \{1\})$ is not open and not closed...

Thank you !

Answer 1

Hint for checking if $S$ is closed or not: consider the point $(1-\frac{1}{k}, \frac{1}{kn}, \ldots, \frac{1}{kn}) \in S$ and take $k \to \infty$.

Hint for checking if $S$ is open or not: pick a point of $S$ and consider any neighborhood of it. Do all points of this neighborhood satisfy the constraints defining $S$?