There are various results, beginning with van der Waerden, on the asymptotic number of integral polynomials with bounded coefficients, whose Galois group is $S_n$.  Some of these results also include $A_n$ in the number of polynomials.  In particular, Rainer Dietmann (*On the distribution of Galois groups*, 2012) shows that the number of polynomials of degree $n$ and height $\leq H$ which do not have Galois group $S_n$ or $A_n$ to be $\ll H^{n-1+e(n)+\varepsilon}$ for a function $e(n)$ which rapidly tends to zero.

On the other hand, Joos Heintz (*On polynomials with symmetric Galois group which are easy to compute*, 1986) shows that the polynomials over a Hilbertian field $k$, considered as points in $k^{n+1}$, which have Galois group $S_n$ are dense.

Is it known or suspected that the polynomials (over $\mathbb{Z}$, $\mathbb{Q}$, or number fields) with a given Galois group $G \not= S_n,A_n$ are discrete?  Is the answer different if instead one takes all polynomials with Galois group different from $S_n,A_n$ or if $A_n$ is dropped from the excluded groups?

**EDIT:**
Certainly it is false over $\mathbb{Z}$ by taking the neighborhood consisting of just a polynomial with Galois group $S_n$.  However, I wonder how close two polynomials with Galois group $\not= S_n,A_n$ can get.