Q:

Examples of $G_\delta$ sets

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 ?

A:

The space of nowhere differentiable continuous functions $f: [0, 1] \to \mathbb{R}$, as a subspace of the space of all continuous functions under the sup-norm topology.  

A:

The set of oracles A (as elements of the product space $2^\omega$) such that $\mathrm P^A\ne\mathrm{NP}^A$. (This also works for nonequality of other oracle complexity classes.)

A:

In the space of all subsets of $\mathbb{N}$ (identified via characteristic functions with $2^{\mathbb{N}}$ and topologized as the product of copies of the discrete 2-point space), the set of infinite subsets of $\mathbb{N}$ is a $G_\delta$ set.  So is the set of those $A\subseteq\mathbb{N}$ for which $\sum_{n\in A}1/n$ diverges.  (The same goes with other sequences in place of $1/n$.)  

