Completeness is a metric property, while closedness is a topological property relative to an ambient space. In complete ambient metric spaces, these two ideas meet: a subspace is complete exactly when it is closed. This result is one of the most useful tools for checking completeness in analysis. :::example[Closed Interval as a Complete Subspace] Since $\mathbb{R}$ is complete and $[0,1]$ is closed in $\mathbb{R}$, the subspace $[0,1]$ is complete. ::: :::example[Open Interval as an Incomplete Subspace] The interval $(0,1)$ is not complete under the usual metric because $1/n$ is a Cauchy sequence in $(0,1)$ that converges to $0\notin(0,1)$. ::: :::theorem[Closed Subspaces of Complete Spaces] Let $(X,d)$ be a complete metric space and let $Y$ be a closed subset of $X$. Then $(Y,d_Y)$ is complete. Conversely, if $Y$ is complete under the induced metric, then $Y$ is closed in $X$. ::: :::proof Given that $Y\subset X$ has the induced metric. To prove the completeness and closedness relation. Suppose $X$ is complete and $Y$ is closed in $X$. Let $\{y_n\}$ be a Cauchy sequence in $Y$. It is also Cauchy in $X$, so it converges to some $x\in X$. Since $Y$ is closed and $y_n\in Y$, the limit $x$ belongs to $Y$. Hence $Y$ is complete. Conversely, suppose $Y$ is complete. Let $x\in\overline{Y}$. Choose a sequence $\{y_n\}$ in $Y$ with $y_n\to x$ in $X$. Then $\{y_n\}$ is Cauchy in $Y$, so it converges to some $y\in Y$. The same convergence holds in $X$. Since limits in metric spaces are unique, $x=y\in Y$. Thus $Y$ is closed. $\square$ ::: :::solved-problem Show that $Y=\{f\in C[0,1]:f(0)=0\}$ is complete under the uniform metric. ::: :::solution The space $C[0,1]$ is complete. If $f_n\in Y$ and $f_n\to f$ uniformly, then $f_n(0)=0$ for every $n$, and so $f(0)=\lim f_n(0)=0$. Hence $Y$ is closed in $C[0,1]$, and therefore complete. $\square$ ::: :::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 Completeness 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/countability-bases-in-metric-spaces button-ghost: Previous Lesson | /functional-analysis/metric-spaces/topological-structure-of-metric-spaces/relative-topology-of-subspaces :::