Borel $\sigma$ algebra on a topological subspace. [duplicate]

Note that if $Y$ is any subspace of $T$, then $B(Y) = \{ A \cap Y : A \in B(T) \}$.

• As $\{ A \cap Y : A \in B(T) \}$ clearly contains all open subsets of $Y$, and is itself a $\sigma$-algebra on $Y$, then $B(Y) \subseteq \{ A \cap Y : A \in B(T) \}$.
• As the inclusion map $i : Y \to T$ is continuous, then $i^{-1} [ A ]$ is a Borel subset of $Y$ for each Borel $A \subseteq T$, but $i^{-1} [ A ] = A \cap Y$, and so $\{ A \cap Y : A \in B(T) \} \subseteq B(Y)$.

If $S \subseteq T$ is Borel, then $A \cap S$ is a Borel subset of $T$ for all Borel $A \subseteq T$, and therefore $\{ A \cap S : A \in B(T) \} = \{ A \in B(T) : A \subseteq S \}$.