Assume that $R$ is a smooth Borel equivalence relation on a Polish space $X$ and $R'$ is a closed sub-equivalence relation; i.e., $R'$ is another Borel equivalence relation on $X$ such that $R'$ is a closed subset of $R\subseteq X\times X$. Is it true that $R'$ is necessarily smooth?
Recall that $R$ is smooth if $X/R$ is a standard Borel space.
Note: $R$ is not necessarily countable and $R'$ is not necessarily a closed subset of $X\times X$.
Since $R$ is smooth there is a finer Polish topology $\tau$ on $X$ such that $R$ is closed as a subset of $(X^2,\tau\times\tau)$, see [1, Corollary 5.4.5]. Since $R'$ is still closed in $(R,\tau\times\tau)$ we see that $R'$ is closed in $(X^2,\tau\times\tau)$, which implies that $R'$ is smooth since every closed equivalence relation on a Polish space is smooth, see for example [1, Proposition 5.4.7] for a reference.
[1] Gao, Su. Invariant descriptive set theory. Chapman and Hall/CRC, 2008.
