When a subset of a metric space is studied with the induced metric, openness and closedness must be interpreted relative to that subspace. A set may be open on a line but not open in the plane. Relative topology explains this dependence on the ambient space and provides the correct form of open and closed sets in subspaces. :::definition[Subspace Metric] Let $(X,d)$ be a metric space and let $Y\subset X$ be nonempty. The restriction of $d$ to $Y\times Y$ is a metric on $Y$, called the $subspace metric$ or $induced metric$. It is given by $d_Y(y_1,y_2)=d(y_1,y_2)$. ::: :::example[Open Set Relative to a Subspace] Let $Y=\{(x,0):x\in\mathbb{R}\}\subset\mathbb{R}^2$ and $Z=\{(x,0):0<x<1\}$. Then $Z$ is open in $Y$ but not open in $\mathbb{R}^2$. ::: :::theorem[Open and Closed Sets in a Subspace] Let $(X,d)$ be a metric space, $Y\subset X$, and $A\subset Y$. Then $A$ is open in $Y$ if and only if $A=G\cap Y$ for some open set $G$ in $X$. Also, $A$ is closed in $Y$ if and only if $A=F\cap Y$ for some closed set $F$ in $X$. ::: :::proof Given that $Y$ is a subspace of $X$. To prove the open and closed set characterizations. First, balls satisfy $S_Y(y,r)=S_X(y,r)\cap Y$. If $A$ is open in $Y$, for each $a\in A$ choose $r_a>0$ such that $S_Y(a,r_a)\subset A$. Let $G=\bigcup_{a\in A}S_X(a,r_a)$. Then $G$ is open in $X$ and $A=G\cap Y$. Conversely, if $A=G\cap Y$ with $G$ open in $X$, then for $a\in A$ there is $r>0$ with $S_X(a,r)\subset G$. Hence $S_Y(a,r)\subset A$, so $A$ is open in $Y$. For closed sets, use complements relative to $Y$. A set $A$ is closed in $Y$ if and only if $Y\setminus A$ is open in $Y$, which is equivalent to $Y\setminus A=G\cap Y$ for some open $G\subset X$. Then $A=(X\setminus G)\cap Y$, with $X\setminus G$ closed. $\square$ ::: :::solved-problem If $A\subset Y\subset X$, describe $\operatorname{cl}_Y A$ in terms of $\operatorname{cl}_X A$. ::: :::solution The closure in the subspace is $$\begin{equation*} \operatorname{cl}_Y A=Y\cap\operatorname{cl}_X A. \end{equation*}$$ This follows because a ball in $Y$ is the intersection of a ball in $X$ with $Y$. $\square$ ::: :::exercise [1] State the main definition from this lesson. [2] State the main theorem from this lesson. [3] Give one example illustrating the theorem. [4] Explain one common mistake related to this topic. ::: :::answer [1] The main definition is the first titled definition in the lesson. [2] The main theorem is the titled theorem proved in the lesson. [3] The examples in the lesson provide standard choices. [4] A common mistake is to ignore the metric or the ambient space when applying the definition. ::: :::faq[Frequently Asked Questions] Q: Why is the ambient metric space important? A: Openness, closedness, closure, and boundary are interpreted relative to the chosen metric space. Q: Can the same set behave differently in different metric spaces? A: Yes. Changing the metric or the ambient space may change topological behaviour. Q: How should students check a definition? A: They should return to balls, closures, and sequences, depending on the context. ::: :::call-to-action[Continue from Relative Topology of Subspaces] subtitle: Review the definitions and proofs before moving through the metric topology sequence. button: Next Lesson | /functional-analysis/metric-spaces/topological-structure-of-metric-spaces/completeness-of-subspaces button-ghost: Previous Lesson | /functional-analysis/metric-spaces/topological-structure-of-metric-spaces/nested-closed-sets-in-complete-spaces :::