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 .

Question

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  .

Tamia Padilla · Accepted Answer

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.)

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

Answered question

Answer & Explanation

New Questions in Pre-Algebra