Borel Sets of a Metric Space: Open and Closed Sets, Points, Open Sets of Subspaces, and Subspaces

In the setting of Routine Facts Used Without Citation: Arithmetic, Maps, Predicativity and Indices, let $(X,d)$ be a metric space, with the topology $\mathcal{T}_{d}$ of Metric Spaces: Balls, the Metric Topology, and the Real and Complex Numbers as Metric Spaces (iii) and the Borel $\sigma$-algebra $\mathcal{B}(X)$.

(i) Every subset of $X$ that is open in $(X,d)$ belongs to $\mathcal{B}(X)$.

(ii) Every subset of $X$ that is closed in $(X,\mathcal{T}_{d})$ belongs to $\mathcal{B}(X)$.

(iii) For every $x\in X$, $\{x\}\in\mathcal{B}(X)$.

(iv) For every $x\in X$ and every real $r\ge0$, the closed ball $\bar{B}_{d}(x,r)$ belongs to $\mathcal{B}(X)$.

Let $Y\subseteq X$, let $d_{Y}=d|_{Y\times Y}$ be the restriction of $d$, a metric on $Y$ by Subsets of a Metric Space: the Restricted Metric, Sequences, Limits in Closed Subsets and Completeness of Closed Subsets (i), and let $\mathcal{B}(Y)=\mathcal{B}(Y,d_{Y})$. Let $\{B\cap Y:B\in\mathcal{B}(X)\}$ be the image of $\mathcal{B}(X)$ under the map $B\mapsto B\cap Y$ from $\mathcal{B}(X)$ to the power set $\mathcal{P}(Y)$ given by Sequences and Maps of Sets: Complements, Traces, Two-Term Sequences, the Disjointing Sequence, and Sequences in a Subset (ii).

(v) A subset $U$ of $Y$ is open in $(Y,d_{Y})$ if and only if $U=V\cap Y$ for some $V$ open in $(X,d)$.

(vi) $B\cap Y\in\mathcal{B}(Y)$ for every $B\in\mathcal{B}(X)$, and $\mathcal{B}(Y)$ is the $\sigma$-algebra on $Y$ generated by $\{B\cap Y:B\in\mathcal{B}(X)\}$.

(vii) If $Y\in\mathcal{B}(X)$, then $\mathcal{B}(Y)=\{B\cap Y:B\in\mathcal{B}(X)\}=\{B\in\mathcal{B}(X):B\subseteq Y\}$.