TITLE: Examples of $G_\delta$ sets
QUESTION [14 upvotes]: Recall that a subset $A$ of a metric space $X$ is a $G_\delta$ subset if it can be written as a countable intersection of open sets. This notion is related to the Baire category theorem. Here are three examples of interesting sets that are 
$G_\delta$ sets.

 The set of continuity points of a function $f:X\rightarrow R$.

 The set of positively recurrent points of a continuous transformation $T: X \rightarrow X$. Recall that a point is recurrent if there exists some sequence $n_i\rightarrow \infty$ such that $T^{n_i}x \rightarrow x$.
 The set of transitive points of a continuous transformation $T: X \rightarrow X$. Recall that a point is transitive if its orbit $\{T^n(x)\}_{n\in Z}$ is dense in $X$.

Question :
Can you provide more examples of interesting $G_\delta$ sets ?

REPLY [2 votes]: Here are a few $G_\delta$-sets related to measure theory.
Let $S$ be a $\sigma$-compact metric space
and denote by ${\cal M}(S)$ the set of Borel positive measures on $S$ endowed
with the weak topology. The following sets are $G_\delta$ sets:

the set of non-atomic measures in ${\cal M}(S)$, 
the set of measures in ${\cal M}(S)$
singular with respect to a given measure,
the set of measures in ${\cal M}(S)$ whose support 
contains a given closed set.

This leads to the following surprising theorem.
Let $U$ be an open subset of ${\bf R}^d$, 
then the set of singular continuous measures is a dense $G_\delta$ subset
of the set of ${\cal M}(U)$. (Lenz, Stollmann 2004).
There are related results in the theory of self-adjoint operators on a separable Hilbert space endowed with the strong resolvent topology. These are $G_\delta$ subsets:

operators with no eigenvalues in a given closed subset of ${\bf R}$,
operators with spectrum containing a given closed subset of ${\bf R}$,
operators such that the spectral measure has no absolutely continuous component on a given open subset of ${\bf R}$.

This leads to the following result.
The set of self-adjoint operators with spectrum $[-1,1]$ 
and with purely singular continuous spectrum is a dense $G_\delta$  subset of the set of self-adjoint operators of norm $\leq 1$ on a separable Hilbert space, endowed with the strong topology.
This is a consequence of the Wonderland theorem of B. Simon, 1993.