Proof That in Euclidean Space There are Sets That are Not Borel Sets

Theorem

In the Euclidean spacethere are some subsets that are not Borel sets.

Proof

Letbe a topological space and let B be the family of Borel sets in

Then
The topological spacehas a countable basis and the cardinal number ofis

Hence

Since card

Hence there are sets inthat are not Borel sets.

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