Take the 2-minute tour ×  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 ?  share|improve this question Natural examples
of Borel class higher than $G_\delta$ or $F_\sigma$ would be
equally interesting. The experts were old masters (like
Kazimierz Kuratowski) about a hundred years ago