Continuous images preserve connectedness, and now we study another preservation principle: adding limit points to a connected set does not break it. Closures and enlargements of connected sets are important because many natural connected sets are most easily described as dense parts of larger spaces. The oscillating sine curve is the standard example: the curve itself is connected, and its closure remains connected even after a whole vertical segment is added. The focus keyword closures and enlargements of connected sets highlights this useful idea. Students should remember that the inclusion $Y\subseteq Z\subseteq\overline{Y}$ is the essential hypothesis. :::theorem[Sets Between a Connected Set and Its Closure] Let $Y$ be a connected subset of a metric space $(X,d)$. If $Z$ satisfies $Y\subseteq Z\subseteq\overline{Y}$, then $Z$ is connected. ::: :::proof Given that $Y$ is connected and $Y\subseteq Z\subseteq\overline{Y}$. To prove that $Z$ is connected. Suppose, if possible, that $Z$ is disconnected. Then there exist nonempty separated subsets $A$ and $B$ of $Z$ such that $Z=A\cup B$. Since $Y\subseteq Z$, we have $Y=(Y\cap A)\cup(Y\cap B)$. If both $Y\cap A$ and $Y\cap B$ are nonempty, then $Y$ is disconnected. This contradicts the connectedness of $Y$. Therefore one of $Y\cap A$ and $Y\cap B$ is empty. Assume, without loss of generality, that $Y\cap B=\varnothing$. Then $Y\subseteq A$. Since $A$ and $B$ are separated in $Z$, no point of $B$ belongs to the closure of $A$ in $Z$. Since $Y\subseteq A$, no point of $B$ belongs to the closure of $Y$ in $Z$. But $Z\subseteq\overline{Y}$, so every point of $Z$, including every point of $B$, belongs to the closure of $Y$. A contradiction. Hence $Z$ is connected. $\square$ ::: :::corollary Let $Y$ be a connected subset of a metric space $(X,d)$. Then $\overline{Y}$ is connected. ::: :::proof Given that $Y$ is connected. To prove that $\overline{Y}$ is connected. We have $Y\subseteq\overline{Y}\subseteq\overline{Y}$. By the theorem on sets between a connected set and its closure, $\overline{Y}$ is connected. $\square$ ::: :::example[Closure of the Oscillating Sine Curve] Let $Y=\{(x,y)\in\mathbb{R}^2:y=\sin(1/x),\ 0<x\leq 1\}$. The set $Y$ is the continuous image of $(0,1]$ under the map $x\mapsto(x,\sin(1/x))$. Since $(0,1]$ is connected and continuous images of connected sets are connected, $Y$ is connected. Its closure is $\overline{Y}=Y\cup\{(0,y):-1\leq y\leq 1\}$. Therefore $\overline{Y}$ is connected. $\square$ ::: :::remark Let $L=\{(0,y):-1\leq y\leq 1\}$. If $Z$ is obtained by keeping $Y$ and adding any subset of $L$, then $Y\subseteq Z\subseteq\overline{Y}$. Therefore $Z$ is connected. This is the central point of closures and enlargements of connected sets. ::: :::solved-problem Let $Y=(0,1)$ as a subset of $\mathbb{R}$. Show that $[0,1)$ is connected by using the theorem on intermediate sets. ::: :::solution Given that $Y=(0,1)$ and $Z=[0,1)$ in $\mathbb{R}$. To prove that $Z$ is connected. The set $Y=(0,1)$ is an interval, so it is connected. Its closure in $\mathbb{R}$ is $\overline{Y}=[0,1]$. We have $Y\subseteq Z\subseteq\overline{Y}$. By the theorem on sets between a connected set and its closure, $Z$ is connected. $\square$ ::: :::exercise [1] State the theorem on sets between a connected set and its closure. [2] Show that the closure of a connected subset is connected. [3] Give an example of a connected set whose closure contains points not in the original set. [4] Explain why adding arbitrary points outside $\overline{Y}$ is not covered by the theorem. ::: :::answer [1] If $Y$ is connected and $Y\subseteq Z\subseteq\overline{Y}$, then $Z$ is connected. [2] Put $Z=\overline{Y}$. [3] The oscillating sine curve has a closure containing the vertical segment at $x=0$. [4] The proof uses density of $Y$ inside $Z$, which may fail outside $\overline{Y}$. ::: :::faq[Frequently Asked Questions] Q: Does every enlargement of a connected set remain connected? A: No. The enlargement must remain inside the closure of the original connected set. Q: Why is the closure of a connected set connected? A: The closure lies between the connected set and its own closure. Q: What is the main geometric meaning? A: Adding limit points to a connected set cannot split it into separated pieces. ::: :::call-to-action[Continue to Unions of Connected Subspaces] subtitle: After closures and enlargements of connected sets, study how connected pieces combine through intersections. button: Next Lesson | /functional-analysis/metric-spaces/connectedness-and-path-structure-in-metric-spaces/unions-of-connected-subspaces button-ghost: Previous Lesson | /functional-analysis/metric-spaces/connectedness-and-path-structure-in-metric-spaces/continuous-images-and-intermediate-values :::