Path-connectedness implies connectedness, and now we study its basic permanence properties. Path-connected images and unions behave much like connected images and unions, but the proof is more constructive because paths can be carried and concatenated. A continuous map sends a path to a path, and a common point allows paths from different pieces to be joined. The focus keyword path-connected images and unions is central here because it gives reusable tools for proving many spaces path-connected. :::theorem[Continuous Image of a Path-Connected Set] Let $(X,d_X)$ and $(Z,d_Z)$ be metric spaces. Let $Y\subseteq X$. If $Y$ is path-connected and $f:(X,d_X)\to(Z,d_Z)$ is continuous, then $f(Y)$ is path-connected. ::: :::proof Given that $Y$ is path-connected and $f:X\to Z$ is continuous. To prove that $f(Y)$ is path-connected. Let $z_1,z_2\in f(Y)$. Then there exist $y_1,y_2\in Y$ such that $f(y_1)=z_1$ and $f(y_2)=z_2$. Since $Y$ is path-connected, there exists a path $g:[0,1]\to Y$ such that $g(0)=y_1$ and $g(1)=y_2$. The composition $f\circ g:[0,1]\to f(Y)$ is continuous. Also, $(f\circ g)(0)=z_1$ and $(f\circ g)(1)=z_2$. Thus $f\circ g$ is a path in $f(Y)$ from $z_1$ to $z_2$. Since $z_1$ and $z_2$ are arbitrary, $f(Y)$ is path-connected. $\square$ ::: :::theorem[Union of Path-Connected Sets with Common Intersection] Let $\mathcal{A}$ be a class of path-connected subsets of a metric space $(X,d)$ such that $\bigcap_{A\in\mathcal{A}}A\neq\varnothing$. Then $B=\bigcup_{A\in\mathcal{A}}A$ is path-connected. ::: :::proof Given that each member of $\mathcal{A}$ is path-connected and that the whole family has a common point. To prove that $B$ is path-connected. Choose $p\in\bigcap_{A\in\mathcal{A}}A$. Let $x,y\in B$. Then there exist $A_x,A_y\in\mathcal{A}$ such that $x\in A_x$ and $y\in A_y$. Since $p\in A_x$ and $A_x$ is path-connected, there exists a path in $A_x$ from $x$ to $p$. Since $p\in A_y$ and $A_y$ is path-connected, there exists a path in $A_y$ from $p$ to $y$. By concatenating these two paths, there exists a path in $B$ from $x$ to $y$. Since $x$ and $y$ are arbitrary, $B$ is path-connected. $\square$ ::: :::example[Image of a Circle Parameterisation] The map $f:\mathbb{R}\to\mathbb{C}$ defined by $f(t)=\cos t+i\sin t$ is continuous. Since $\mathbb{R}$ is path-connected, its image $f(\mathbb{R})=\{z\in\mathbb{C}:|z|=1\}$ is path-connected. $\square$ ::: :::solved-problem Show that the union of all line segments from the origin to points of the unit circle is path-connected. ::: :::solution Given that $B$ is the union of all line segments from the origin to points of the unit circle. To prove that $B$ is path-connected. Each line segment is path-connected because it is convex. All these line segments contain the common point $(0,0)$. Therefore, by the union theorem for path-connected sets with common intersection, their union $B$ is path-connected. $\square$ ::: :::exercise [1] Prove that a continuous image of a path-connected set is path-connected. [2] State the common-intersection theorem for path-connected unions. [3] Give an example of a path-connected union using line segments. [4] Explain why a common point is useful in joining paths. ::: :::answer [1] Compose a path in the domain with the continuous map. [2] A family of path-connected sets with nonempty total intersection has path-connected union. [3] All line segments from the origin to points on a circle have path-connected union. [4] Paths from each piece can be concatenated through the common point. ::: :::faq[Frequently Asked Questions] Q: Does every continuous image of a path-connected set remain connected? A: Yes, it remains path-connected, and therefore connected. Q: Can a union of path-connected sets fail to be path-connected? A: Yes, if the pieces do not overlap suitably. Q: Why does composition preserve paths? A: A path is a continuous map from $[0,1]$, and the composition of continuous maps is continuous. ::: :::call-to-action[Continue to Open Connected Sets in the Plane] subtitle: Use path-connected images and unions to prove that open connected subsets of the plane are path-connected. button: Next Lesson | /functional-analysis/metric-spaces/connectedness-and-path-structure-in-metric-spaces/open-connected-sets-in-the-plane button-ghost: Previous Lesson | /functional-analysis/metric-spaces/connectedness-and-path-structure-in-metric-spaces/path-connectedness-and-ordinary-connectedness :::