Symmetric matrices with distinct eigenvalues form a connected set

Idea. The spectral theorem gives a parametrisation of $\Omega$ by an orthonormal basis of eigenvectors and an ordered list of eigenvalues. Both parameter spaces are connected (provided we use $SO(n)$ and not $O(n)$).

Step 1: the parametrisation. Let $\Delta=\{\lambda\in\mathbb{R}^n:\lambda_1>\dots>\lambda_n\}$, an open convex set, hence connected. Define
$$\Phi:SO(n)\times\Delta\to\mathrm{Sym}_n,\qquad\Phi(Q,\lambda)=Q\operatorname{diag}(\lambda)Q^T.$$
$\Phi$ is continuous, and $\Phi(Q,\lambda)\in\Omega$ since its eigenvalues are the $\lambda_i$.

Step 2: surjectivity onto $\Omega$. Let $S\in\Omega$ with eigenvalues $\lambda_1>\dots>\lambda_n$. By the spectral theorem $S=Q\Lambda Q^T$ with $Q\in O(n)$ (columns = unit eigenvectors in the order of the eigenvalues). If $\det Q=-1$, replace $Q$ by $Q'=QD$ with $D=\operatorname{diag}(-1,1,\dots,1)$, i.e. flip the sign of the first eigenvector. Then $\det Q'=1$ and $Q'\Lambda Q'^T=QD\Lambda DQ^T=Q\Lambda Q^T=S$ because $D$ commutes with the diagonal matrix $\Lambda$. So $S=\Phi(Q',\lambda)$.

Step 3: connectedness of the source. $SO(n)$ is connected (standard: every $U\in SO(n)$ is orthogonally conjugate to a block diagonal matrix with rotation blocks $R_{\theta_j}$ and $1$'s, and $t\mapsto$ the same matrix with $\theta_j$ replaced by $t\theta_j$ is a path to the identity; the conjugating matrix can be taken in $SO(n)$ by flipping the sign of a $+1$ eigenvector or of one vector in a plane, which only changes $\theta_j$ to $-\theta_j$). A product of connected spaces is connected.

Step 4: conclusion. $\Omega=\Phi(SO(n)\times\Delta)$ is a continuous image of a connected space, hence connected.

Remark. $\Omega$ is open and dense in $\mathrm{Sym}_n$. Its complement (matrices with a repeated eigenvalue) has codimension $2$ in $\mathrm{Sym}_n$ (a double eigenvalue costs $3$ conditions but frees... one finds $\dim=\frac{n(n+1)}2-2$), which is why removing it does not disconnect $\mathrm{Sym}_n$. For $n=2$, $\mathrm{Sym}_2\cong\mathbb{R}^3$ and the complement is the line of scalar matrices.

Lesson. Parametrise a set by connected pieces and use that continuous images of connected sets are connected; use $SO(n)$ rather than $O(n)$ when connectedness is needed.