Let $M$ be real non-singular symmetric $n \times n$ matrix with $p$ positive and $n-p$ negative eigenvalues. What is the topology of the space of such matrices?
For a trivial case $n=1$ the matrix is an ordinary number (which is also its eigenvalue), there are two 1-dimensional spaces with a trivial topology, one for negative and one for positive eigenvalue.
For $n=2$, if eigenvalues have the same sign, space of such matrices is 3-dimensional space of trivial topology, but if they have different signs, then space of such matrices is isomorphic to $\mathbb R^2 \times S^1$.
How to find the topology for a general case of $p$ positive and $n-p$ negative eigenvalues?