Invariance of domain theorem tells us that if a subset V of <mrow class="MJX-TeXAtom

EnvivyEvoxys6 2022-07-05 Answered
Invariance of domain theorem tells us that if a subset V of R n is homeomorphic to an open subset of R n , then V must be open itself.
Question: If a subset V of Rn is homeomorphic to a Borel subset of R n , must V be Borel ?
Recall Borel( R n ) is defined to be the σ-algebra generated by the topology of R n .
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Answers (1)

Tamia Padilla
Answered 2022-07-06 Author has 16 answers
The answer to the question is yes.
If B R n is Borel and f : R n R n is continuous such that f | B is injective, then the image f ( B ) R n is Borel.

We also have the measurable analog of the Invariance of Domain:

2. If B R n is Borel and f : R n R n is Borel such that f | B is injective, then the image f ( B ) R n is Borel and f | B : B f ( B ) is a Borel isomorphism.

(In general one can replace R n with a standard Borel space.)
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