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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.

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  • $\begingroup$ My thoughts: a) $det A \ne 0$ is open and dense, open since matrices that are "almost" non-singular are non-singluar, and dense since all nonsingular matrices have determinants close to a number in $R$. b) Trace of A is rational is a dense set since all the rational traces are close to values in $R^n$. c) Entries of A are not integers is not open but is dense. $\endgroup$ – bowen.jane Sep 23 '16 at 17:22
  • $\begingroup$ Are this really the definitions you have? Because that's awful. What does sufficiently near mean? For example: the set a) is open since it is a preimage of a open set under a continous function. Are you familiar with this concepts? $\endgroup$ – Maik Pickl Sep 23 '16 at 18:10
  • $\begingroup$ @MaikPickl yes, unfortunately those are the definitions my book has. I have worked with set theory a bit before so I have some idea about open sets (though not preimage...), but I'm not sure how to apply it to sets of matrices. I have also never used the "dense" property, so I'm still unsure of what that means. $\endgroup$ – bowen.jane Sep 23 '16 at 18:24

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