# On Borel $\sigma$-algebra of subset as a subspace

Let $$S$$ be a subset of the set of real numbers $$\mathbb{R}$$, let $$\mathcal{B}$$ be the Borel $$\sigma$$-algebra generated by all open subsets of $$\mathbb{R}$$. Consider $$S$$ as a topological space endowed with the subspace topology (i.e., the topology inherited from $$\mathbb{R}$$), and let $${\mathcal{B}}_S$$ denote the Borel $$\sigma$$-algebra generated by all open subsets of S. Is it true that $${\mathcal{B}}_S\subset \mathcal{B}$$ ? (Here, the set $$S$$ is not necessarily a Borel set in $$\mathbb{R}$$.) I think, in general, this is not true. However, I could not find a counter-example.

• math.stackexchange.com/a/2693750/659976 this might help – Edgar Jaber May 26 at 13:51
• The answers in this reference show that if $S$ is not a Borel set in $\mathbb{R}$ then the inclusin $\mathcal{B}_S\subset \mathcal{B}$, in general, is NOT true. Am I right? – serenus May 26 at 14:29

You are right in that it is not true in general. Let $$S$$ be a subset of $$\mathbb{R}$$ which is not Borel measurable. But since $$S$$ is open with the subspace topology, $$S \in \mathcal{B}_S$$ but not in $$\mathcal{B}$$.

However, if $$S$$ is open, then this is true. Let $$S$$ be open in $$\mathbb{R}$$ and let $$\mathcal{B}_S$$ be Borel sigma algebra on the subspace topology. Then since all subsets of $$S$$ which are open w.r.t. $$S$$ are also open w.r.t. $$\mathbb{R}$$, each open subset of $$S$$ also lies in $$\mathcal{B}$$ which is thus a sigma algebra containing all open subsets of $$S$$ and thus $$\mathcal{B}_S \subseteq \mathcal{B}$$.

If $$S$$ is closed, the proof is similar.

• I think that, more generally, when $S$ is a Borel set in $\mathbb{R}$, then we have $\mathcal{B}_S\subset \mathcal{B}$. – serenus May 26 at 16:36
• Yes, that's true. Let $S$ be Borel, then every open subset of $S$ is of the form $S \cap U$ for some $U \subseteq \mathbb{R}$ open. Hence it is also Borel, thus all sets which are open in $S$ lie in $\mathcal{B}$ and thus $\mathcal{B}_S$ does as well. – G. Chiusole May 26 at 16:54