$\newcommand{\br}{\\}$ $\newcommand{\R}{\mathbb{R}}$ $\newcommand{\Q}{\mathbb{Q}}$ $\newcommand{\Z}{\mathbb{Z}}$ $\newcommand{\N}{\mathbb{N}}$ $\newcommand{\C}{\mathbb{C}}$ $\newcommand{\P}{\mathbb{P}}$ $\newcommand{\F}{\mathbb{F}}$ $\newcommand{\L}{\mathcal{L}}$ $\newcommand{\spa}[1]{\text{span}(#1)}$ $\newcommand{\dist}[1]{\text{dist}(#1)}$ $\newcommand{\max}[1]{\text{max}(#1)}$ $\newcommand{\min}[1]{\text{min}(#1)}$ $\newcommand{\supr}[1]{\text{sup}(#1)}$ $\newcommand{\infi}[1]{\text{inf}(#1)}$ $\newcommand{\ite}[1]{\text{int}(#1)}$ $\newcommand{\ext}[1]{\text{ext}(#1)}$ $\newcommand{\bdry}[1]{\partial #1}$ $\newcommand{\argmax}[1]{\underset{#1}{\text{argmax }}}$ $\newcommand{\argmin}[1]{\underset{#1}{\text{argmin }}}$ $\newcommand{\set}[1]{\left\{#1\right\}}$ $\newcommand{\emptyset}{\varnothing}$ $\newcommand{\tilde}{\text{~}}$ $\newcommand{\otherwise}{\text{ otherwise }}$ $\newcommand{\if}{\text{ if }}$ $\newcommand{\proj}{\text{proj}}$ $\newcommand{\union}{\cup}$ $\newcommand{\intercept}{\cap}$ $\newcommand{\abs}[1]{\left| #1 \right|}$ $\newcommand{\norm}[1]{\left\lVert#1\right\rVert}$ $\newcommand{\pare}[1]{\left(#1\right)}$ $\newcommand{\brac}[1]{\left[#1\right]}$ $\newcommand{\t}[1]{\text{ #1 }}$ $\newcommand{\head}{\text H}$ $\newcommand{\tail}{\text T}$ $\newcommand{\d}{\text d}$ $\newcommand{\limu}[2]{\underset{#1 \to #2}\lim}$ $\newcommand{\der}[2]{\frac{\d #1}{\d #2}}$ $\newcommand{\derw}[2]{\frac{\d #1^2}{\d^2 #2}}$ $\newcommand{\pder}[2]{\frac{\partial #1}{\partial #2}}$ $\newcommand{\pderw}[2]{\frac{\partial^2 #1}{\partial #2^2}}$ $\newcommand{\pderws}[3]{\frac{\partial^2 #1}{\partial #2 \partial #3}}$ $\newcommand{\inv}[1]{{#1}^{-1}}$ $\newcommand{\inner}[2]{\langle #1, #2 \rangle}$ $\newcommand{\nullity}[1]{\text{nullity}(#1)}$ $\newcommand{\rank}[1]{\text{rank }#1}$ $\newcommand{\nullspace}[1]{\mathcal{N}\pare{#1}}$ $\newcommand{\range}[1]{\mathcal{R}\pare{#1}}$ $\newcommand{\var}[1]{\text{var}\pare{#1}}$ $\newcommand{\cov}[1]{\text{cov}(#1)}$ $\newcommand{\cov}[2]{\text{cov}\pare{#1, #2}}$ $\newcommand{\tr}[1]{\text{tr}(#1)}$ $\newcommand{\oto}{\text{ one-to-one }}$ $\newcommand{\ot}{\text{ onto }}$ $\newcommand{\ceil}[1]{\lceil#1\rceil}$ $\newcommand{\floor}[1]{\lfloor#1\rfloor}$ $\newcommand{\Re}[1]{\text{Re}(#1)}$ $\newcommand{\Im}[1]{\text{Im}(#1)}$ $\newcommand{\dom}[1]{\text{dom}(#1)}$ $\newcommand{\fnext}[1]{\overset{\sim}{#1}}$ $\newcommand{\transpose}[1]{{#1}^{\text{T}}}$ $\newcommand{\b}[1]{\boldsymbol{#1}}$ $\newcommand{\None}[1]{}$ $\newcommand{\Vcw}[2]{\begin{bmatrix} #1 \br #2 \end{bmatrix}}$ $\newcommand{\Vce}[3]{\begin{bmatrix} #1 \br #2 \br #3 \end{bmatrix}}$ $\newcommand{\Vcr}[4]{\begin{bmatrix} #1 \br #2 \br #3 \br #4 \end{bmatrix}}$ $\newcommand{\Vct}[5]{\begin{bmatrix} #1 \br #2 \br #3 \br #4 \br #5 \end{bmatrix}}$ $\newcommand{\Vcy}[6]{\begin{bmatrix} #1 \br #2 \br #3 \br #4 \br #5 \br #6 \end{bmatrix}}$ $\newcommand{\Vcu}[7]{\begin{bmatrix} #1 \br #2 \br #3 \br #4 \br #5 \br #6 \br #7 \end{bmatrix}}$ $\newcommand{\vcw}[2]{\begin{matrix} #1 \br #2 \end{matrix}}$ $\newcommand{\vce}[3]{\begin{matrix} #1 \br #2 \br #3 \end{matrix}}$ $\newcommand{\vcr}[4]{\begin{matrix} #1 \br #2 \br #3 \br #4 \end{matrix}}$ $\newcommand{\vct}[5]{\begin{matrix} #1 \br #2 \br #3 \br #4 \br #5 \end{matrix}}$ $\newcommand{\vcy}[6]{\begin{matrix} #1 \br #2 \br #3 \br #4 \br #5 \br #6 \end{matrix}}$ $\newcommand{\vcu}[7]{\begin{matrix} #1 \br #2 \br #3 \br #4 \br #5 \br #6 \br #7 \end{matrix}}$ $\newcommand{\Mqw}[2]{\begin{bmatrix} #1 & #2 \end{bmatrix}}$ $\newcommand{\Mqe}[3]{\begin{bmatrix} #1 & #2 & #3 \end{bmatrix}}$ $\newcommand{\Mqr}[4]{\begin{bmatrix} #1 & #2 & #3 & #4 \end{bmatrix}}$ $\newcommand{\Mqt}[5]{\begin{bmatrix} #1 & #2 & #3 & #4 & #5 \end{bmatrix}}$ $\newcommand{\Mwq}[2]{\begin{bmatrix} #1 \br #2 \end{bmatrix}}$ $\newcommand{\Meq}[3]{\begin{bmatrix} #1 \br #2 \br #3 \end{bmatrix}}$ $\newcommand{\Mrq}[4]{\begin{bmatrix} #1 \br #2 \br #3 \br #4 \end{bmatrix}}$ $\newcommand{\Mtq}[5]{\begin{bmatrix} #1 \br #2 \br #3 \br #4 \br #5 \end{bmatrix}}$ $\newcommand{\Mqw}[2]{\begin{bmatrix} #1 & #2 \end{bmatrix}}$ $\newcommand{\Mwq}[2]{\begin{bmatrix} #1 \br #2 \end{bmatrix}}$ $\newcommand{\Mww}[4]{\begin{bmatrix} #1 & #2 \br #3 & #4 \end{bmatrix}}$ $\newcommand{\Mqe}[3]{\begin{bmatrix} #1 & #2 & #3 \end{bmatrix}}$ $\newcommand{\Meq}[3]{\begin{bmatrix} #1 \br #2 \br #3 \end{bmatrix}}$ $\newcommand{\Mwe}[6]{\begin{bmatrix} #1 & #2 & #3\br #4 & #5 & #6 \end{bmatrix}}$ $\newcommand{\Mew}[6]{\begin{bmatrix} #1 & #2 \br #3 & #4 \br #5 & #6 \end{bmatrix}}$ $\newcommand{\Mee}[9]{\begin{bmatrix} #1 & #2 & #3 \br #4 & #5 & #6 \br #7 & #8 & #9 \end{bmatrix}}$
Definition: Relative Topology

Let $(X, d) $ be a metric space, let $Y$ be a subset of $X$, and let $E$ be a subset of $Y $. We say that $E $ is relatively open with respect to $Y$ if it is open in the metric subspace $(Y, d|_{Y \times Y}) $. Similarly, we say that $E $ is relatively closed with respect to $Y $ if it is closed in the metric space $(Y, d| _{Y \times Y}) $.

Proposition 1.3.4

Let $(X, d) $ be a metric space, let $Y$ be a subset of $X$, and let $E $ be a subset of $Y $.

(a) $E $ is relatively open with respect to $Y $ if and only if $E = E \cap Y $ for some set $V \subseteq X $ which is open in $X $.

(b) $E $ is relatively closed with respect to $Y $ if and only if $E = K \cap Y $ for some set $K \subseteq X $ which is closed in $X$.

Proof
  </span>
</span>
<span class="proof__expand"><a>[expand]</a></span>

(Sufficiency)

Suppose that $E $ is relatively open with respect to $Y$. Then, $E$ is open in the metric space $(Y, d|_{ Y \times Y }) $. Thus,

$$\forall x \in E, \exists r_x > 0, B_{(Y, d|_{ Y \times Y })} (x, r_x) \subseteq E $$

Now consider the set

$$V := {\bigcup}_{ x \in E } B_{(X, d)} (x, r_x) \subseteq X$$

Since the union of open sets is still an open set, $V$ is open.

Now we prove that $E = V \cap Y $.

$\forall x \in E, x B_{(X, d)} (x, r_x) \subseteq V \cap Y$.

$\therefore E \subseteq V \cap Y $.

Fix $y \in V \cap Y. y \in V \implies \exists x \in E, y \in B_{(X, d)}(x, r_x)$.

$y \in Y \implies y \in B_{(Y, d|_{ Y \times Y })} (x, r_x)$.

By the definition of $r_x$, this means that $y \in E $.

$\therefore E = V \cap Y $. (Q.E.D.)

(Necessity)

Suppose $E = V \cap Y$ for some open set $V$; want to show that $E$ is relatively open with respect to $Y$. That is,

Fix $x \in E$, want to show that $x$ is an interior point of $E$ in the metric space $(Y, d|_{Y \times Y})$.

$x \in E \implies x \in V$

$V\text{ is open in }X \implies \exists r > 0, B_{(X, d)} (x, r) \subseteq V$

$E = V \cap Y \implies B_{(X,d)}(x, r) \cap Y \subseteq E$.

$B_{(X,d)}(x, r) \cap Y = B_{(Y, d|_{ Y \times Y })} (x, r) \implies B(Y, d|_{(Y, d|_{ Y \times Y })}) \subseteq E$.

Thus $x$ is an interior point of $E$ in the metric space $(Y, d|_{ Y \times Y })$.