Invariance of domain theorem tells us that if a subset $V$ of ${\mathbb{R}}^{n}$ is homeomorphic to an open subset of ${\mathbb{R}}^{n}$, then $V$ must be open itself.

Question: If a subset $V$ of Rn is homeomorphic to a Borel subset of ${\mathbb{R}}^{n}$, must $V$ be Borel ?

Recall Borel(${\mathbb{R}}^{n}$) is defined to be the σ-algebra generated by the topology of ${\mathbb{R}}^{n}$.

Question: If a subset $V$ of Rn is homeomorphic to a Borel subset of ${\mathbb{R}}^{n}$, must $V$ be Borel ?

Recall Borel(${\mathbb{R}}^{n}$) is defined to be the σ-algebra generated by the topology of ${\mathbb{R}}^{n}$.