Each of the following properties define a subset of real n x n matrices. Which of these sets are open and/or dense in the $L(R^n)$? Give a brief reason in each case.
a) $det A \ne 0$
b) Trace A is rational
c) Entries of A are not integers
d) $3 \le det A \lt 4$
e) $-1\lt |\lambda|\lt 1$ for every eigenvalue $\lambda$
f) A has no real eigenvalues
g) Each real eigenvalue of A has multiplicity one
My textbook lists the following definitions for open and dense: a set U is open if whenever $x\in U$ there is an open ball about x contained in U, or that any point sufficiently near x is in U. A set is dense if there are points in U arbitrarily close to each point in $R^n$. From these definitions, I have a few guesses about some of the a)-g) above, but I'm not sure all the guesses are right, and some of the parts I still don't have guesses for.