TITLE: Topological properties of symmetric positive definite matrices
QUESTION [8 upvotes]: Let $S$ be the set of all symmetric positive definite matrices of size $n\times n$. Which of the following statements are true?   
(a) $S$ is closed in $\mathbb{M}_n(\mathbb{R})$.
 (b) $S$ is connected in $\mathbb{M}_n(\mathbb{R})$.
 (c) $S$ is compact in $\mathbb{M}_n(\mathbb{R})$.        
Only the option (a) & (b) are right. I guess that it is not bounded so (c) is not true. Am I correct?

REPLY [2 votes]: As La Belle Noiseuse said, $S$ is a convex set; $S$ is also open in a vector space of dimension $n(n+1)/2$; thus $S$ is homeomorphic to the open ball of $\mathbb{R}^{n(n+1)/2}$ for an arbitrary norm.
EDIT for @ Sahiba Arora. The property "$A$ is heomorphic to a bounded set" gives no information about the fact that $A$ is bounded or not. Indeed, the open ball of $\mathbb{R}^{n(n+1)/2}$ is homeorphic to $\mathbb{R}^{n(n+1)/2}$.
Here $S$ is not closed (consider the sequence $(\dfrac{1}{k}I_n)_k$) and is not bounded (consider the sequence $(kI_n)_k$).